text:
On dynamics of infinitely generated contractive mapping systems on \$p\mathbb{Z}_p\$

 Speaker 杨静桦 博士 DateTime 2019年01月18日（周五）上午10:00-11:30 Place 六号楼二楼报告厅 Abstract In this talk,   I also show some results on limit sets of semigroups generated by infinitely many contractive maps with open tiling conditions.  It is shown that there exists a shift transformation on the limit set of G and the shift transformation  is ergodic with respect to the Haar measure on p Zp, if G satisfies the open tiling conditions. Furthermore, we can generalize p-adic Khinchin's Theorem and p-adic Lochs' Theorem  to any infinitely generated semigroup by use of the ergodicity of the shift transformation.


text:
2015
07-17

Mad Veterinarian puzzles have a mad veterinarian, who has developed several machines that can transform an animal into one or more animals and back again. The puzzle is then to determine if it is possible to change one collection of animals into another by applying the machines in some order (forward or reverse). For example:

Machine A turns one ant into one beaver.
Machine B turns one beaver into one ant, one beaver and one cougar.
Machine C turns one cougar into one ant and one beaver.

Can we convert a beaver and a cougar into 3 ants?

Can we convert one ant into 2 ants? NO

These puzzles have the properties that:

1. In forward mode, each machine converts one animal of a given species into a finite, non-empty collection of animals from the species in the puzzle.
2. Each machine can operate in reverse.
3. There is one machine for each species in the puzzle and that machine (in forward mode) takes as input one animal of that species.

Write a program to find the shortest solution (if any) to Mad Veterinarian puzzles. For this problem we will restrict to Mad Veterinarian puzzles with exactly three machines, A, B, C.

The first line of input contains a single integer P, (1<= P <= 1000 ), which is the number of data sets that follow. Each data set consists of several lines of input. Each data set should be processed identically and independently.

The first line of each data set consists of two decimal integers separated by a single space. The first integer is the data set number. The second integer is the number, N, of puzzle questions. The next three input lines contain the descriptions of machines A, B and C in that order. Each machine description line consists of three decimal integers separated by spaces giving the number of animals of type a, b and c output for one input animal. The following N lines give the puzzle questions for the Mad Veterinarian puzzle. Each contains seven decimal digits separated by single spaces: the puzzle number, the three starting animal counts for animals a, b and c followed by the three desired ending animal counts for animals a, b and c.

The first line of input contains a single integer P, (1<= P <= 1000 ), which is the number of data sets that follow. Each data set consists of several lines of input. Each data set should be processed identically and independently.

The first line of each data set consists of two decimal integers separated by a single space. The first integer is the data set number. The second integer is the number, N, of puzzle questions. The next three input lines contain the descriptions of machines A, B and C in that order. Each machine description line consists of three decimal integers separated by spaces giving the number of animals of type a, b and c output for one input animal. The following N lines give the puzzle questions for the Mad Veterinarian puzzle. Each contains seven decimal digits separated by single spaces: the puzzle number, the three starting animal counts for animals a, b and c followed by the three desired ending animal counts for animals a, b and c.

2
1 2
0 1 0
1 1 1
1 1 0
1 0 1 1 3 0 0
2 1 0 0 2 0 0
2 2
0 3 4
0 0 5
0 0 3
1 2 0 0 0 0 5
2 2 0 0 0 0 4

1 2
1 3 Caa
2 NO SOLUTION
2 2
1 NO SOLUTION
2 25 AcBcccBccBcccAccBccBcccBc

10以上就不要了我会乱说?队列只要开1000我会乱说?

→_→

#include<cstdio>
#include<cstring>
using namespace std;
bool vis[12][12][12];
int tras[4][4];
struct node
{
int a[4];
int dis;
}st,end,q[1000];
char path[1000];
int fa[1000];
char h[]="ZABCabc";
int num;
void print(int k)
{
if(!k) return;
print(fa[k]);
putchar(path[k]);
}
void bfs()
{
node f,r;
int rear=0,front=0;
st.dis=0;
q[rear++]=st;
vis[st.a[1]][st.a[2]][st.a[3]]=1;
while(front<rear)
{
f=q[front];
f.dis++;
for(int i=1;i<=3;i++)
{
r=f;
if(r.a[i])
{
r.a[i]--;
for(int j=1;j<=3;j++)
{
r.a[j]+=tras[i][j];
}
if(r.a[1]>10||r.a[2]>10||r.a[3]>10) continue;
if(!vis[r.a[1]][r.a[2]][r.a[3]])
{
fa[rear]=front;
path[rear]=h[i];
if(r.a[1]==end.a[1]&&r.a[2]==end.a[2]&&r.a[3]==end.a[3]) {printf("%d %d ",num,r.dis);print(rear);return;}
vis[r.a[1]][r.a[2]][r.a[3]]=1;
q[rear++]=r;
}
}
}
for(int i=4;i<=6;i++)
{
r=f;
int ok=1;
for(int j=1;j<=3;j++)
{
if(r.a[j]<tras[i-3][j]) ok=0;
}
if(ok)
{
for(int j=1;j<=3;j++)
{
r.a[j]-=tras[i-3][j];
}
r.a[i-3]++;
if(r.a[1]>10||r.a[2]>10||r.a[3]>10) continue;
if(!vis[r.a[1]][r.a[2]][r.a[3]])
{
path[rear]=h[i];
fa[rear]=front;
if(r.a[1]==end.a[1]&&r.a[2]==end.a[2]&&r.a[3]==end.a[3]) {printf("%d %d ",num,r.dis);print(rear);return;}
vis[r.a[1]][r.a[2]][r.a[3]]=1;
q[rear++]=r;
}
}
}
front++;
}
printf("%d NO SOLUTION",num);
}
int main()
{
int p,casq,n;
scanf("%d",&p);
while(p--)
{
scanf("%d %d",&casq,&n);
printf("%d %d\n",casq,n);
for(int i=1;i<=3;i++)
{
for(int j=1;j<=3;j++)
scanf("%d",&tras[i][j]);
}
for(int i=1;i<=n;i++)
{
scanf("%d %d %d %d %d %d %d",&num,&st.a[1],&st.a[2],&st.a[3],&end.a[1],&end.a[2],&end.a[3]);
if(st.a[1]==end.a[1]&&st.a[2]==end.a[2]&&st.a[3]==end.a[3])
{
printf("%d 0\n",num);
continue;
}
memset(vis,0,sizeof(vis));
bfs();
puts("");
}
}
return 0;
}

format: plain


text:
# How to find an integer set, s.t. the sums of at most 3 elements are all distinct?

How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different.

Example with $|A|=3$: Out of the set $A := \{1,7,11\}$ follow 19 sums 1,2,3,7,8,9,11,12,13,14,15,18,19,21,22,23,25,29,33 which are all distinct!

Is there an algorithm to construct such sets for a given number of elements $n=|A|$ with the smallest maximum possible?

-
This is what I figured out so far: Such sets are known as $B_k[1]$ sets. k is the number of elements in the sum. Therefore im interested in $B_3[1]$ sets. Sets with distinct sums of only two elements $B_2[1]$ are also known as Sidon sets. –  Shannon Jul 19 '14 at 20:25
The first place to turn for this kind of question is Richard Guy's book, Unsolved Problems In Number Theory. Unfortunately, I don't have it handy right now. Also, it won't be as up-to-date as Tony Huynh's answer, but it will give some context and history and references. –  Gerry Myerson Jul 20 '14 at 1:08

There is an algorithm: look at all $n$-element subsets of $S:=\{1,2,3,\dots,4^{n-1}\}$. There is at least one $B_3[1]$ subset among them, namely $\{1,4,4^2,\dots,4^{n-1}\}$. So list all $n$-element $B_3[1]$ subsets of $S$, and pick the one with the smallest maximum.

-

Note that there is a bit of a discrepancy between $B_3[1]$ sets and your notion. For $B_3[1]$ sets, one considers sums with exactly 3 elements, instead of at most 3 elements. However, I do not think this will make a big difference. Basically the main point is that we can do much better than GH from MO's answer by taking a random subset of integers and using the probabilistic method. Indeed, this was carried out by Erdős–Rényi for Sidon sets and claimed (but not proven) for $B_3[1]$ sets. The first correct proof for $B_3[1]$ sets is attributed to Vu in this thesis of Carlos Vinuesa del Río. That thesis also has the current state of the art for your problem. See Theorem 2.1.2. Specializing that result we have the following theorem.

Theorem. For any $\epsilon > 0$, there is an infinite $B_3[1]$ sequence $A$ such that $A(x) \gg x^{\frac{1}{5} - \epsilon}$.

Here $A(x)$ is the number of elements of $A$ that are at most $x$.

-
Note that the OP only asked if there is an algorithm to find a suitable set with given cardinality and smallest maximum. The possible answers to this question are "yes", "no", "undecidable". With all that, I appreciate your response which contains far more mathematics. –  GH from MO Jul 19 '14 at 23:00
Yes, I absolutely agree that you completely answered the question as asked. I +1ed it. –  Tony Huynh Jul 19 '14 at 23:21
Isn't it enough to add $0$ to the set to get a $B_3[1]$ set? (Bearing in mind I have not looked up what $B_3[1]$ means...) –  Bruno Le Floch Jul 20 '14 at 3:51

The probabilistic method shows the existence of an infinite $B_h[1]$ sequence of positive integers $A$ with $A(x)>x^{1/(2h-1)+o(1)}$. The proof is easy for $h=2$ but involved for $h\ge 3$. In particular it gives the exponent $1/5$ for $h=3$.

Ruzsa improved the exponent $1/3$ to $\sqrt 2-1$ for $h=2$ and very recently Cilleruelo, using a new method, improved the exponent $1/(2h-1)$ to $\sqrt{(h-1)^2+1}-(h-1)$ for any $h\ge 3$. http://www.sciencedirect.com/science/article/pii/S000187081400022X

Furthermore, for $h=2$ this method provides an explicit $B_2[1]$ sequence $A$ with $A(x)>x^{\sqrt 2-1+o(1)}$.

-


text:
## CryptoDB

### Paper: Argument of knowledge of a bounded error

Authors: Vadym Fedyukovych URL: http://eprint.iacr.org/2008/359 Search ePrint Search Google A protocol is introduced to show knowledge of a codeword of Goppa code and Goppa polynomial. Protocol does not disclosure any useful information about the codeword and polynomial coefficients. A related protocol is introduced to show Hamming weight of an error is below a threshold. Protocol does not disclosure codeword and weight of the error. Verifier only uses commitments to codeword components and coefficients while testing validity of statements. Both protocols are honest verifier zero knowledge.
##### BibTeX
@misc{eprint-2008-18036,
title={Argument of knowledge of a bounded error},
booktitle={IACR Eprint archive},
keywords={cryptographic protocols / Goppa code, argument of knowledge, Schwartz-Zippel lemma, zero knowledge},
url={http://eprint.iacr.org/2008/359},
note={unpublished vf@unity.net 14109 received 18 Aug 2008},


text:
# Why are combinators important in lambda calculus?

I just recently learned a little about the lambda calculus, from the brief intro in the text Programming Language Pragmatics and this outstanding 4-video sequence from Adam Doupé. Basically I learned the gist of lambda calculus, $\alpha$-conversion and $\beta$-reduction, a bit of Church numerals, addition/subtraction, and the $Y$ combinator.

In this I learned the concept of a combinator, i.e. that it is an abstraction with no free variables. Simple concept. And there seemed to be this idea I kept reading between the lines, that we can/should turn non-combinator abstractions into combinators by wrapping them in abstractions that bind the free variables. For example, we can take the abstraction $(\lambda x. xy)$ and wrap it in $(\lambda y. \lambda x. xy)$ to bind the $y$.

But it was never really explained why it is important or useful to prefer combinators.

I'm assuming its because combinators allow us to easily combine expressions (surprise surprise) without the need of any externally-supplied variables. So in other words we could evaluate an arbitrary-length series of combinator applications without having to provide any initial conditions ahead of time. Which to me sounds like the equivalent in programming of avoiding global variables so as to avoid looking up values outside the current scope, in order to make it easier to understand the program and reduce errors. It also seems to align with the idea of a "pure functional" language that has no variables and no side-effects, which would imply such a language is effectively a series of combinator applications.

Is that correct?

The word "combinator" has some connotations that you don't seem to be intending here and sometimes a stricter definition. Another term for the definition you gave is a closed term. The opposite is an open term. The programming language equivalent of an open term would be an expression referring to a variable that's simply not in scope. Not just not in the current scope, but not in any containing scopes either. Most languages simply don't allow this. We can model let expressions that locally bind a name with lambda abstraction and application. In particular, let x = E in B becomes $(\lambda x.B)E$. The top-level, or global, scope can then be thought of conceptually as just a short-hand for wrapping your program in a bunch of let expressions. The point being, even global names are bound. The analogy to an open term, in this context, would be a term with one or more completely undefined variables in it.

Now clearly a closed term is self-contained which is useful. Further, for the purposes of computation, we never need to consider anything other than closed terms (assuming we start with nothing but closed terms). More precisely, for computation we usually care about weak (head) normal form which is the normal form for call-by-value(/name) reduction. An important aspect of these reduction strategies is they never reduce under a lambda. As such, any instance of $\beta$-reduction using these reduction strategies will only involve closed terms and will only ever produce a closed term assuming the initial term was closed. So, for the purposes of programming and computation closed terms are all that matters. Optimizing compilers, though, manipulate programs and often do perform reduction under a lambda and as such do need to deal with open terms. Proof assistants also need to deal with open terms as they usually need to reduce to normal form to compare lambda terms for equality. (Actually, they can often avoid going all the way to normal form.)

The focus on "combinators" isn't driven by software engineering concerns. It's simply a fact that for many purposes you only need to consider closed terms. The theory and formal manipulation of open terms is quite a bit hairier than for closed terms, so it is useful to limit to closed terms when possible.

• Great answer thanks. This is along the lines of what I was trying to get at but I couldn't word it as clearly as you did. Is it useful to study lambda calculus for purposes other than influencing functional programming? – Dave Jan 20 '17 at 0:35
• I should clarify my question but I can't edit it -- I'm sure it is useful to study it for other reasons, but I'm not sure what reasons there are for studying it. – Dave Jan 20 '17 at 0:42
• @Dave The original purpose for the lambda calculus in the 1930s was understanding mathematical logic. There are deep connections various typed lambda calculi and category theory, and from there to physics as covered in Physics, Topology, Logic and Computation: A Rosetta Stone. There are applications to linguistics. I'm sure there are more. The connection to category theory allows it to be transplanted into many other fields, e.g. graph theory, though the significance isn't always clear. – Derek Elkins left SE Jan 20 '17 at 1:29
• Ok great. I knew about the basic origins and that it theoretically can be a math foundation, but not how far it reaches into theory today. Thanks. – Dave Jan 20 '17 at 1:40

A combinator is a lambda expression with no free variables.

So, λx.x is a combinator but λx.y is not a combinator.

Consider the following:

(λxy.xy)(λx.y) = (λy.xy)[x := λx.y] = (λy.(λx.y)y)

Notice how the final lambda expression is incorrect because the previously free variable y in λx.y has become bound. That's why when you're combining lambda expressions with free variables you have to follow renaming rules to ensure that the expression that results has the same meaning it had at the start.

Hence, this seems to me to be one of the reasons combinators are important:

When substituting with combinators your reduced expression will never change in meaning.

P.S. I'm now learning this stuff so I could be wrong.

• Someone on IRC freenode in the #haskell-beginners channel (Cale) had this to say: "The more important thing is that combinators, having no remaining free variables, are the terms which are in some sense intrinsically meaningful -- there's nothing left to be determined from context about what is being expressed. It's the same as the difference between a proof following from assumptions, and a proof of something which follows entirely from the rules of the logic.". – Dwayne Crooks Jan 19 '17 at 22:12
• This is essentially what I would have answered if you hadn't, with the minor difference that I would have said "renaming" rather than "disambiguation". Different perspective on the same idea. – Peter Taylor Jan 19 '17 at 22:37
• I came across the disambiguation term in the Wikipedia article on lambda calculus but renaming is definitely a clearer term. – Dwayne Crooks Jan 19 '17 at 22:43
• Right, I understood that you have to rename before substituting, to avoid free variable capture. Really good summary insight -- I like the focus on meaning there and in your comment. It seems to sum up the importance. Thanks. – Dave Jan 20 '17 at 1:20
• You've basically got it, in that one of the key problems with lambda calculus is that a naive implementation of beta reduction can suffer from name capture. (By the way, this is also a problem in real-world compilers, where "inlining" a function may require renaming all variables inside it depending on how the compiler represents variables.) It's also interesting to know that a programming language doesn't need variables! By the way, it also has practical applications. See research.microsoft.com/en-us/um/people/simonpj/papers/… – Pseudonym Jan 20 '17 at 3:34


text:
# GATE Papers >> CSE >> 2016 >> Question No 119

Question No. 119

Let X be the number of distinct 16-bit integers in 2's complement representation. Let Y be the number of distinct 16-bit integers in sign magnitude representation. Then X-Y is ________.

##### Answer : 1.0 : 1.0

Solution of Question No 119 of GATE 2016 CSE Paper

Number of bits n = 16

In sign 2's complement  representation the range is

= -215 to + (215 - 1)

= -32768 to + 32767

$\therefore$         X = 65536

In sign magnitude represntation the range is

= (215 - 1) to + (215 - 1)

= -32767 to + 32767

$\therefore$         Y = 65536

Hence

X - Y = 1


text:
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# IEEE Transactions on Circuits and Systems II: Express Briefs

## Filter Results

Displaying Results 1 - 16 of 16

Publication Year: 2013, Page(s): C1
| PDF (38 KB)
• ### IEEE Transactions on Circuits and Systems—II: Express Briefs publication information

Publication Year: 2013, Page(s): C2
| PDF (135 KB)
• ### A K-Band UWB Low-Noise CMOS Mixer With Bleeding Path $G_{m}$-Boosting Technique

Publication Year: 2013, Page(s):117 - 121
Cited by:  Papers (6)
| | PDF (602 KB) | HTML

A low-noise ultrawideband CMOS mixer circuit for the down conversion of a K-band signal is presented. A bleeding path gm-boosting technique is proposed, which reduces the effect of noise sources of the proposed mixer. The measurement results show that the minimum noise figure is 3.61 dB at 24 GHz when consuming 2.6 mW at a 1.5-V supply voltage. The results also show a conversion gain of... View full abstract»

• ### A 0.6-V +4 dBm IIP3 LC Folded Cascode CMOS LNA With gm Linearization

Publication Year: 2013, Page(s):122 - 126
Cited by:  Papers (12)
| | PDF (619 KB) | HTML

This brief presents the design guidelines of LNAs under low-supply-voltage condition with respect to linearity and demonstrates an LNA that has excellent performance even with an extremely low supply voltage. Under a low supply voltage, the drain conductance nonlinearity, which can be ignored in a high supply voltage, is important, as well as the transconductance nonlinearity. Therefore, this brie... View full abstract»

• ### A 96-GHz Oscillator by High-Q Differential Transmission Line loaded with Complementary Split-Ring Resonator in 65-nm CMOS

Publication Year: 2013, Page(s):127 - 131
Cited by:  Papers (10)
| | PDF (979 KB) | HTML

A 96-GHz CMOS oscillator is demonstrated in this brief with the use of a high-Q metamaterial resonator. The proposed metamaterial resonator is constructed by a differential transmission line (T-line) loaded with complementary split-ring resonator engraved on the T-line. A negative real part of permittivity, i.e., , is observed near the resonance frequency, which introduces a sharp stopband and, th... View full abstract»

• ### Design of a wide-tuning-range lithium tantalate low-phase-noise voltage-controlled oscillator

Publication Year: 2013, Page(s):132 - 136
Cited by:  Papers (5)
| | PDF (692 KB) | HTML

In this brief, a lithium tantalate crystal with a low capacitance ratio and a wide tuning range is used to design a Butler common-base wide-tuning-range low-phase-noise oscillator considering a reasonable compromise between seeking high loaded quality factor and maintaining oscillator output power. The formulation of is derived from analysis of the Butler oscillator circuit and calculated with MAT... View full abstract»

• ### Phase Noise Modeling for Integrated PLLs in FMCW Radar

Publication Year: 2013, Page(s):137 - 141
Cited by:  Papers (6)
| | PDF (305 KB) | HTML

The effect of frequency modulation (FM) on the phase noise spectrum of a charge pump phase-locked loop (PLL) for frequency-modulated continuous-wave (FMCW) radar is investigated. The static phase error is calculated as a function of frequency ramp slope and basic PLL design parameters. The phase noise spectrum under FM is deduced from the spectrum in the steady state. Design guidelines for integra... View full abstract»

• ### A 10-bit 25-MS/s 1.25-mW Pipelined ADC With a Semidigital Gm-Based Amplifier

Publication Year: 2013, Page(s):142 - 146
Cited by:  Papers (6)
| | PDF (746 KB) | HTML

A semidigital Gm-based amplifier is proposed for a low-power pipelined analog-to-digital converter (ADC). The amplifier performs a class-AB operation by smoothly changing between a comparator-like semidigital driver and a continuous-time high-gain amplifier according to the input voltage difference. A 10-bit pipelined ADC with 2.5-bit/stage architecture is implemented in a 0.13- CMOS. The ADC cons... View full abstract»

• ### Thermal Noise Analysis of a Programmable-Gain Switched-Capacitor Amplifier With Input Offset Cancellation

Publication Year: 2013, Page(s):147 - 151
Cited by:  Papers (3)
| | PDF (387 KB) | HTML

In this brief, we analyze the thermal noise in a switched-capacitor (SC) amplifier with input offset cancellation. We quantify the increase in thermal noise due to the extra phase required for offset sampling. We show that this noise penalty increases with the SC amplifier gain. A closed-form expression is obtained for the output RMS noise voltage for the special case where the amplifier is design... View full abstract»

• ### Capacitor Multiplier With Wide Dynamic Range and Large Multiplication Factor for Filter Applications

Publication Year: 2013, Page(s):152 - 156
Cited by:  Papers (3)
| | PDF (934 KB) | HTML

A capacitor multiplier realization based on current-voltage conversion with a high-performance voltage follower is presented. The architecture has a large multiplication factor of up to 10 000 or larger compared with current-mode multipliers, and a wide dynamic range compared with voltage-mode capacitor multipliers. It offers higher accuracy than voltage-mode multipliers as the design is based on ... View full abstract»

• ### Two-Phase Self-Assisted Zero-Voltage Switching DC–DC Converter

Publication Year: 2013, Page(s):157 - 161
Cited by:  Papers (2)
| | PDF (735 KB) | HTML

This paper proposes a two-phase self-assisted zero-voltage switching (ZVS) converter suitable for fully monolithic implementation. The proposed topology utilizes an auxiliary inductor connected between the two branches of a two-phase converter to achieve ZVS conditions. The proposed converter helps reduce the output voltage ripples of the standard ZVS topology. Furthermore, the auxiliary inductor ... View full abstract»

• ### Low-Complexity Parallel QPP Interleaver Based on Permutation Patterns

Publication Year: 2013, Page(s):162 - 166
Cited by:  Papers (4)
| | PDF (413 KB) | HTML

In this brief, we present how parallel-interleaved addresses generated by a quadratic permutation polynomial (QPP) interleaver are related to each other and propose a low-complexity parallel QPP interleaver based on the relationship. While a conventional parallel turbo decoder employs a number of interleavers as many as the parallel factor, the proposed method, which benefits from the arithmetic r... View full abstract»

• ### Design of Two-Dimensional Zero-Phase FIR Digital Filter by McClellan Transformation and Interval Global Optimization

Publication Year: 2013, Page(s):167 - 171
Cited by:  Papers (2)
| | PDF (574 KB) | HTML

The McClellan transformation is widely used to design the 2-D FIR filters. It is an essential task to determine the transform coefficients so that the square errors are minimized. This issue can be viewed as a nonlinear optimization problem with constraint conditions. This brief proposed the interval analysis (IA) to solve this nonlinear optimization problem. There are two distinguishing features ... View full abstract»

• ### Dynamic Braess's Paradox in Complex Communication Networks

Publication Year: 2013, Page(s):172 - 176
Cited by:  Papers (5)
| | PDF (119 KB) | HTML

Braess's paradox shows a counterintuitive scenario where more resources may cause worse traffic performance when all users pursue their personal optimum. Previous study on this topic mainly focused on the specific traffic volume to show the existence of such a paradox. For real-world complex communication networks, on the contrary, the traffic volume changes from time to time. Therefore, it is imp... View full abstract»

• ### IEEE Circuits and Systems Society Information

Publication Year: 2013, Page(s): C3
| PDF (117 KB)
• ### IEEE Transactions on Circuits and Systems—II: Express Briefs information for authors

Publication Year: 2013, Page(s): C4
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## Aims & Scope

Part I will now contain regular papers focusing on all matters related to fundamental theory, applications, analog and digital signal processing. Part II will report on the latest significant results across all of these topic areas.

Full Aims & Scope

## Meet Our Editors

Editor-in-Chief

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Dept. of Electronic and Information Engineering
Hong Kong Polytechnic University
Hunghom, Hong Kong
cktse@ieee.org


text:
# Length contraction

In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer. This contraction (more formally called Lorentz contraction or Lorentz–FitzGerald contraction) is usually only noticeable at a substantial fraction of the speed of light; the contraction is only in the direction parallel to the direction in which the observed body is travelling. This effect is negligible at everyday speeds, and can be ignored for all regular purposes. Only at greater speeds does it become important. At a speed of 13,400,000 m/s (30 million mph, 0.0447c), the length is 99.9% of the length at rest; at a speed of 42,300,000 m/s (95 million mph, 0.141c), the length is still 99%. As the magnitude of the velocity approaches the speed of light, the effect becomes dominant, as can be seen from the formula:

$L=\frac{L_{0}}{\gamma(v)}=L_{0}\sqrt{1-v^{2}/c^{2}}$

where

L0 is the proper length (the length of the object in its rest frame),
L is the length observed by an observer in relative motion with respect to the object,
v is the relative velocity between the observer and the moving object,
c is the speed of light,

and the Lorentz factor, γ(v), is defined as

$\gamma (v) \equiv \frac{1}{\sqrt{1-v^2/c^2}} \$.

In this equation it is assumed that the object is parallel with its line of movement. For the observer in relative movement, the length of the object is measured by subtracting the simultaneously measured distances of both ends of the object. For more general conversions, see the Lorentz transformations. An observer at rest viewing an object travelling very close to the speed of light would observe the length of the object in the direction of motion as very near zero.

## History

Length contraction was postulated by George FitzGerald (1889) and Hendrik Antoon Lorentz (1892) to explain the negative outcome of the Michelson-Morley experiment and to rescue the hypothesis of the stationary aether (Lorentz–FitzGerald contraction hypothesis).[1][2] Although both FitzGerald and Lorentz alluded to the fact that electrostatic fields in motion were deformed ("Heaviside-Ellipsoid" after Oliver Heaviside, who derived this deformation from electromagnetic theory in 1888), it was considered an Ad hoc hypothesis, because at this time there was no sufficient reason to assume that intermolecular forces behave the same way as electromagnetic ones. In 1897 Joseph Larmor developed a model in which all forces are considered as of electromagnetic origin, and length contraction appeared to be a direct consequence of this model. Yet it was shown by Henri Poincaré (1905) that electromagnetic forces alone cannot explain the electron's stability. So he had to introduce another ad-hoc hypothesis: non-electric binding forces (Poincaré stresses) that ensure the Electron's stability, give a dynamical explanation for length contraction, and thus hide the motion of the stationary aether. Eventually, Albert Einstein (1905) was the first who completely removed the ad-hoc character from the contraction hypothesis, by demonstrating that this contraction was no dynamical effect in the aether, but rather a kinematic effect due to the change in the notions of space, time and simultaneity brought about by special relativity. Einstein's view was further elaborated by Hermann Minkowski and others, who demonstrated the geometrical meaning of all relativistic effects in spacetime. So length contraction is not of dynamic, but kinematic origin.[3][4][5]

## Basis in relativity

First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects, where "object" simply means a distance with endpoints that are always mutually at rest, i.e., that are at rest in the same inertial frame of reference. If the relative velocity between an observer (or his measuring instruments) and the observed object is zero, then the proper length $L_0$ of the object can simply be determined by directly superposing a measuring rod. However, if the relative velocity > 0, then one can proceed as follows: The observer installs a row of clocks that either are synchronized a) by exchanging light signals according to the Poincaré-Einstein synchronization, or b) by "slow clock transport", that is, one clock is transported along the row of clocks in the limit of vanishing transport velocity. Now, when the synchronization process is finished, the object is moved along the clock row and every clock stores the exact time when the left or the right end of the object passes by. After that, the observer only has to look after the position of a clock A that stored the time when the left end of the object was passing by, and a clock B at which the right end of the object was passing by at the same time. It's clear that distance AB is equal to length $L$ of the moving object.

Symmetry of Lorentz contraction: Three blue rods are at rest in S, and three red rods in S'. At the instant when the left ends of A and D attain the same position on the axis of x, the lengths of the rods shall be compared. In S the simultaneous positions of the left side of A and the right side of C are more distant than those of D and F. While in S' the simultaneous positions of the left side of D and the right side of F are more distant than those of A and C.

Thus the definition of simultaneity is crucial for measuring the length of moving objects. In Newtonian mechanics, simultaneity is absolute and therefore $L$ and $L_0$ are always equal. Yet in relativity theory the constancy of light velocity in all inertial frames in connection with the relativity of simultaneity destroys this equality. So if an observer in one frame claims to have measured the object's endpoints simultaneously, the observers in all other inertial frames will argue that the object's endpoints were not measured simultaneously. The deviation between the measurements in all inertial frames is given by the Lorentz transformation. As the result of this transformation (see Derivation), the proper length remains unchanged and always denotes the greatest length of an object, yet the length of the same object as measured in another inertial frame is shorter than the proper length. This contraction only occurs in the line of motion, and can be represented by the following relation (where $v$ is the relative velocity and $c$ the speed of light)

$L = L_0 \cdot \sqrt{1-\frac{v^2}{c^2}}.$

For example, a train at rest in S' and a station at rest in S with relative velocity of $v = 0{.}8c$ are given. In S' a rod with proper length $L_0^{'}=30\ \mathrm{cm}$ is located, so its contracted length $L'$ in S is given by:

$L = L_0^{'} \cdot \sqrt{1-\frac{v^2}{c^2}} = 18\ \mathrm{cm}.$

Then the rod will be thrown out of the train in S' and will come to rest at the station in S. Its length has to be measured again according to the methods given above, and now the proper length $L_0 = 30\ \mathrm{cm}$ will be measured in S (the rod has become larger in that system), while in S' the rod is in motion and therefore its length is contracted (the rod has become smaller in that system):

$L' = L_0 \cdot \sqrt{1-\frac{v^2}{c^2}} = 18\ \mathrm{cm}.$

Thus, as it is required by the principle of relativity (according to which the laws of nature must assume the same form in all inertial reference frames), length contraction is symmetrical: If the rod is at rest in the train, it has its proper length in S' and its length is contracted in S. However, if the rod comes to rest relative to the station, it has its proper length in S and its length is contracted in S'.

## Derivation

### Lorentz transformation

Length contraction can simply be derived from the Lorentz transformation as it was shown, among many others, by Max Born:[6]

In an inertial reference frame S', $x_{1}^{'}$ and $x_{2}^{'}$ shall denote the endpoints for an object of length $L_{0}^{'}$ at rest in this system. The coordinates in S' are connected to those in S by the Lorentz transformations as follows:

$x_{1}^{'}=\frac{x_{1}-vt_{1}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$    and    $x_{2}^{'}=\frac{x_{2}-vt_{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.$

As this object is moving in S, its length $L$ has to be measured according to the above convention by determining the simultaneous positions of its endpoints, so we have to put $t_{1}=t_{2}\,$. Because $L=x_{2}-x_{1}\,$ and $L_{0}^{'}=x_{2}^{'}-x_{1}^{'}$, we obtain

(1) $L_{0}^{'}=\frac{L}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.$

Thus the length as measured in S is given by

(2) $L=L_{0}^{'}\cdot\sqrt{1-\frac{v^{2}}{c^{2}}}.$

According to the relativity principle, objects that are at rest in S have to be contracted in S' as well. For this case the Lorentz transformation is as follows:

$x_{1}=\frac{x_{1}^{'}+vt_{1}^{'}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$     and    $x_{2}=\frac{x_{2}^{'}+vt_{2}^{'}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.$

By the requirement of simultaneity $t_{1}^{'}=t_{2}^{'}\,$ and by putting $L_{0}=x_{2}-x_{1}\$ and $L^{'}=x_{2}^{'}-x_{1}^{'}$, we actually obtain:

(3) $L_{0}=\frac{L^{'}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.$

Thus its length as measured in S' is given by:

(4) $L^{'}=L_{0}\cdot\sqrt{1-\frac{v^{2}}{c^{2}}}.$

So (1), (3) give the proper length when the contracted length is known, and (2), (4) give the contracted length when the proper length is known.

### Time dilation

Length contraction can also be derived from time dilation,[7] according to which the rate of a single "moving" clock (indicating its proper time $T_0$) is lower with respect to two synchronized "resting" clocks (indicating $T$). Time dilation was experimentally confirmed multiple times, and is represented by the relation:

$T=\frac{T_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$.

Suppose a rod of proper length $L_0$ at rest in $S$ and a clock at rest in $S'$ are moving along each other. The respective travel times of the clock between the rod's endpoints are given by $T=L_{0}/v$ in $S$ and $T'_{0}=L'/v$ in $S'$, thus $L_{0}=Tv$ and $L'=T'_{0}v$. By inserting the time dilation formula, the ratio between those lengths is:

$\frac{L'}{L_{0}}=\frac{T'_{0}v}{Tv}=\sqrt{1-\frac{v^{2}}{c^{2}}}$

Therefore, the length measured in $S'$ is given by

$L'=L_{0}\sqrt{1-\frac{v^{2}}{c^{2}}}$.

So the effect that the moving clock indicates a lower travel time in $S$ due to time dilation, is interpreted in $S'$ as due to length contraction of the moving rod. Likewise, if the clock were at rest in $S$ and the rod in $S'$, the above procedure would give

$L=L'_{0}\sqrt{1-\frac{v^{2}}{c^{2}}}$.

## Geometrical representation

Minkowski diagram and length contraction. In S all events parallel to the axis of x are simultaneous, while in S' all events parallel to the axis of x' are simultaneous.

The Lorentz transformation geometrically corresponds to a rotation in four-dimensional spacetime, and it can be illustrated by a Minkowski diagram: If a rod at rest in S' is given, then its endpoints are located upon the ct' axis and the axis parallel to it. In this frame the simultaneous (parallel to the axis of x') positions of the endpoints are O and B, thus the proper length is given by OB. But in S the simultaneous (parallel to the axis of x) positions are O and A, thus the contracted length is given by OA. On the other hand, if another rod is at rest in S, then its endpoints are located upon the ct axis and the axis parallel to it. In this frame the simultaneous (parallel to the axis of x) positions of the endpoints are O and D, thus the proper length is given by OD. But in S' the simultaneous (parallel to the axis of x') positions are O and C, thus the contracted length is given by OC.

Left: a rotated cuboid in three-dimensional euclidean space E3. The cross section is longer in the direction of the rotation than it was before the rotation. Right: the world slab of a moving thin plate in Minkowski spacetime (with one spatial dimension suppressed) E1,2, which is a boosted cuboid. The cross section is thinner in the direction of the boost than it was before the boost. In both cases, the transverse directions are unaffected and the three planes meeting at each corner of the cuboids are mutually orthogonal (in the sense of E1,2 at right, and in the sense of E3 at left).

Additional geometrical considerations show, that length contraction can be regarded as a trigonometric phenomenon, with analogy to parallel slices through a cuboid before and after a rotation in E3 (see left half figure at the right). This is the Euclidean analog of boosting a cuboid in E1,2. In the latter case, however, we can interpret the boosted cuboid as the world slab of a moving plate.

In special relativity, Poincaré transformations are a class of affine transformations which can be characterized as the transformations between alternative Cartesian coordinate charts on Minkowski spacetime corresponding to alternative states of inertial motion (and different choices of an origin). Lorentz transformations are Poincaré transformations which are linear transformations (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the Lorentz group forms the isotropy group of the self-isometries of the spacetime) which are played by rotations in euclidean geometry. Indeed, special relativity largely comes down to studying a kind of noneuclidean trigonometry in Minkowski spacetime, as suggested by the following table:[A 1]

## Experimental verifications

Any observer co-moving with the observed object cannot measure the object's contraction, because he can judge himself and the object as at rest in the same inertial frame in accordance with the principle of relativity (as it was demonstrated by the Trouton-Rankine experiment). So Length contraction cannot be measured in the object's rest frame, but only in a frame in which the observed object is in motion. In addition, even in such a non-co-moving frame, direct experimental confirmations of Lorentz contraction are hard to achieve, because at the current state of technology, objects of considerable extension cannot be accelerated to relativistic speeds. And the only objects traveling with the speed required are atomic particles, yet whose spatial extensions are too small to allow a direct measurement of contraction.

However, there are indirect confirmations of this effect in a non-co-moving frame. It was in fact the negative result of a famous experiment, that required the introduction of Lorentz contraction: the Michelson-Morley experiment (and later also the Kennedy–Thorndike experiment). In special relativity its explanation is as follows: In its rest frame the interferometer can be regarded as at rest in accordance with the relativity principle, so the propagation time of light is the same in all directions. Although in a frame in which the interferometer is in motion, the transverse beam must traverse a longer, diagonal path with respect to the non-moving frame thus making its travel time longer, the factor by which the longitudinal beam would be delayed by taking times L/(c-v) & L/(c+v) for the forward and reverse trips respectively is even longer. Therefore, in the longitudinal direction the interferometer is supposed to be contracted, in order to restore the equality of both travel times times in accordance with the negative experimental result(s). Thus the two-way speed of light remains constant and the round trip propagation time along perpendicular arms of the interferometer is independent of its motion & orientation.

Other indirect confirmations are: Heavy ions that are spherical when at rest should assume the form of "pancakes" or flat disks when traveling nearly at the speed of light. And in fact, the results obtained from particle collisions can only be explained, when the increased nucleon density due to Lorentz contraction is considered.[8][9][10][11] Another confirmation is the increased ionization ability of electrically charged particles in motion. According to pre-relativistic physics the ability should decrease at high speed, however, the Lorentz contraction of the Coulomb field leads to an increase of the electrical field strength normal to the line of motion, which leads to the actually observed increase of the ionization ability.[12] Lorentz contraction is also necessary to understand the function of free-electron lasers. Relativistic electrons were injected into an undulator, so that synchrotron radiation is generated. In the proper frame of the electrons, the undulator is contracted which leads to an increased radiation frequency. Additionally, to find out the frequency as measured in the laboratory frame, one has to apply the relativistic Doppler effect. So, only with the aid of Lorentz contraction and the rel. Doppler effect, the extremely small wavelength of undulator radiation can be explained.[13][14] Another example is the observed lifetime of muons in motion and thus their range of action, which is much higher than that of muons at low velocities. In the proper frame of the atmosphere, this is explained by the time dilation of the moving muons. However, in the proper frame of the muons their lifetime is unchanged, but the atmosphere is contracted so that even their small range is sufficient to reach the surface of earth.[12]

## Reality of Lorentz contraction

Another issue that is sometimes discussed concerns the question whether this contraction is "real" or "apparent". However, this problem only stems from terminology, as our common language attributes different meanings to both of them. Yet, whatever terminology is chosen, in physics the measurement and the consequences of length contraction with respect to any reference frame are clearly and unambiguously defined in the way stated above.[15]

In 1911 Vladimir Varićak asserted that length contraction is "real" according to Lorentz, while it is "apparent or subjective" according to Einstein. Einstein replied:

The author unjustifiably stated a difference of Lorentz's view and that of mine concerning the physical facts. The question as to whether the Lorentz contraction really exists or not is misleading. It doesn't "really" exist, in so far as it doesn't exist for a comoving observer; though it "really" exists, i.e. in such a way that it could be demonstrated in principle by physical means by a non-comoving observer.[16]
—Albert Einstein, 1911

Due to superficial application of the contraction formula some paradoxes can occur. For examples see the Ladder paradox or Bell's spaceship paradox. However, those paradoxes can simply be solved by a correct application of relativity of simultaneity. Another famous paradox is the Ehrenfest paradox, which proves that the concept of rigid bodies is not compatible with relativity, and reduces the applicability of Born rigidity. It also shows that for a co-rotating observer the geometry is in fact non-euclidean.

## Visual effects

Length contraction refers to measurements of position made at simultaneous times according to a coordinate system. This could naively lead to a thinking that if one could take a picture of a fast moving object, that the image would show the object contracted in the direction of motion. However, it is important to realize that such visual effects are completely different measurements, as such a photograph is taken from a distance, while length contraction can only directly be measured at the exact location of the object's endpoints. In 1959 Roger Penrose and James Terrell published papers demonstrating that length contraction instead actually shows up as elongation or even a rotation in a photographic image.[17][18][19] This kind of visual rotation effect is called Penrose-Terrell rotation.[20]

## Notes

1. ^
Three plane trigonometries
Trigonometry Circular Parabolic Hyperbolic
Kleinian Geometry euclidean plane Galilean plane Minkowski plane
Symbol E2 E0,1 E1,1
Quadratic form positive definite degenerate non-degenerate but indefinite
Isometry group E(2) E(0,1) E(1,1)
Isotropy group SO(2) SO(0,1) SO(1,1)
type of isotropy rotations shears boosts
Cayley algebra complex numbers dual numbers split-complex numbers
ε2 -1 0 1
Spacetime interpretation none Newtonian spacetime Minkowski spacetime
slope tan φ = m tanp φ = u tanh φ = v
"cosine" cos φ = (1+m2)-1/2 cosp φ = 1 cosh φ = (1-v2)-1/2
"sine" sin φ = m (1+m2)-1/2 sinp φ = u sinh φ = v (1-v2)-1/2
"secant" sec φ = (1+m2)1/2 secp φ = 1 sech φ = (1-v2)1/2
"cosecant" csc φ = m−1 (1+m2)1/2 cscp φ = u−1 csch φ = v−1 (1-v2)1/2

## References

1. ^ FitzGerald, George Francis (1889), "The Ether and the Earth's Atmosphere", Science 13 (328): 390, Bibcode:1889Sci....13..390F, doi:10.1126/science.ns-13.328.390, PMID 17819387
2. ^ Lorentz, Hendrik Antoon (1892), "The Relative Motion of the Earth and the Aether", Zittingsverlag Akad. V. Wet. 1: 74–79
3. ^ Einstein, Albert (1905), "Zur Elektrodynamik bewegter Körper", Annalen der Physik 322 (10): 891–921, Bibcode:1905AnP...322..891E, doi:10.1002/andp.19053221004. See also: English translation.
4. ^ Janssen, Michel H. P. (2007), "Drawing the line between kinematics and dynamics in special relativity", Symposium on Time and Relativity: 1–76
5. ^ Torretti, Roberto (2006), "Can science advance effectively through philosophical criticism and reflection?", [no journal title given]: 1–84
6. ^ Born, Max (1964), Einstein's Theory of Relativity, Dover Publications, ISBN 0-486-60769-0
7. ^ David Halliday, Robert Resnick, Jearl Walker (2010), Fundamentals of Physics, Chapters 33-37, John Wiley & Son, pp. 1032f, ISBN 0470547944
8. ^ The Physics of RHIC
9. ^ Relativistic heavy ion collisions
10. ^
11. ^ Simon Hands, The Phase Diagram of QCD, Contemp. Phys. 42:209-225, 2001, arXiv:physics/0105022
12. ^ a b Sexl, Roman & Schmidt, Herbert K. (1979), Raum-Zeit-Relativität, Braunschweig: Vieweg, ISBN 3-528-17236-3


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Correct Answer Your Answer; 1: x 2 + 3 x + 2: Solution The factored form of the trinomial will take the form of (x + ?)(x + ?) Front coefficient is 1. Select your answer from the popup menu. Emphasis is on the use of mixed numbers. Even if you know a dozen = 12, the relationship. To factor trinomials with a leading factor greater than 1, I use the "magic number" method. What You'll Need A set of four matching solution cards per player. 5 Lesson 3: Finding Factors, Sums, and Differences pg. ) When converting a single unit, such as converting from centimeters to meters, the given should also be a single unit as well. multiplying polynomials worksheet 1 answer key generated on lbartman. Click on the links below to view sample pages. Factoring Binomials Calculator Type in the binomial and select factor. Ex: You may move. When two shapes are similar, their corresponding angles will be the same. A frequently-used procedure in algebra is the process of combining like terms. For example, 42 = 6· 7. In this file, students will practice solving one and two step equations. Find the least common multiple of two whole numbers less than or equal to 12. Fun Math Links; Distance Learning with Google Classroom; Algebra 1 Homework; Algebra 1 Review Packets; Algebra 2 Homework; How Access the Online Big Ideas Textbook; Quadratic Project Rubric; Math Binder Set Up; Student Success Strategies. And at the end they have a snazzy picture! 1/10/18 - A new Clue Activity for Solving Quadratics by Factoring. Our online polynomial trivia quizzes can be adapted to suit your requirements for taking some of the top polynomial quizzes. 2𝑥2=9𝑥+5 A 3 𝑥2−16=0 A 6 3𝑥2+2=7𝑥 B 5 8 𝑥2−5𝑥=36 C 1. atomic structure worksheet answer Price and Picture. Roman Numeral Challenge. This Factors and Multiples Game is a fun way for students to identify factors and multiples of given numbers in a jeopardy game style format. r G 7Mia AdoeE qw 5i at Ih I oIgn1f Jiundiit0ee nA7l1g SeFb YrLa4 N1M. Perhaps a key factor in such decisions, beyond titles or responsibilities, is the health and age of those staffers themselves, NBA general managers told ESPN. Factor: a number or algebraic expression by which another is exactly divisible. Directions: Factor the following polynomials. Today we are going to practice factoring & solving (or finding zeros) of higher degree polynomials. Which list is made up of multiples of 4?. Factoring Maze Instructions: Start out the maze by factoring every polynomial. Write the genotype for a homozygous dominant striped squash. Factoring Trinomials (a = 1) Date_____ Period____ Factor each completely. She carried ones in the places I’ve shown. If you call your show "news" you need to be objective. With this worksheet generator, you can make customizable worksheets for the distributive property and factoring. Intermediate Algebra Skill Factoring the Sum or Difference of Cubes Factor each completely. If you can't factor it write "prime" Factor the common factor out of each expression. ) Do not forget to apply significant figures to your final answer. We hope that you find exactly what you need for your home or classroom!. Graph a linear function. Use the settings and options below to generate the type of questions you want to include on the worksheet. 7 + 14x 17. Have Fun Teaching Resources such as worksheets, activities, workbooks, songs, and videos are perfect for teaching. Math Expert. They simplify expressions containing like terms. Scale Drawing. 5x + 35 16. 6 = 2 * 3 You can divide 6 by 2 or by 3. 5 is a multiple of 20. Choose a scale large enough to produce a good drawing, but small enough to fit on your paper. Then determine which number is a perfect number. On this page you will find: a complete list of all of our math worksheets, lessons, math homework, and quizzes. We also worked through the Factoring Trinomials by Grouping lesson, and if we’d had more time, we would have played the Old Poly game. Quadratics Unit Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. In your drawing, show the different sizes of the planets and where they are located. xx 6 or 10 5. When someone asks, "Who owns the customer's change-of-address experience?" they always know the answer. 50x2 372 9. But to do the job properly we need the highest common factor, including any variables. ) 3rd and 4th Grades. 1 – 3 Stress experiences can be emotionally or physiologically challenging and activate stress responses and adaptive processes to regain homeostasis. Trinomials - Undoing FOIL 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. 18, 24 and 30 are all multiples of three. Step #3: Perform factor by grouping on. I can factor using difference of squares. Henry Ford took inspiration from the past, saw opportunities for the future, and believed in technology as a force for improving people's lives. Tennis Center Re-opens Friday, May 8th. Select several values for M and try to establish what the flow chart does. Click "Show Answer" underneath the problem to see the answer. I ask students to think with their. #N#Answer pages are only available to subscribers of EnchantedLearning. With this worksheet generator, you can make customizable worksheets for the distributive property and factoring. x2 -49 -95. Interpreting data on bar graphs and picture graphs can be more fun with these worksheets. Braingenie is the Web's most comprehensive math and science practice site. Factor Trees Complete the factor tree for each number to find the prime factors. Since factoring is a product of factors, we first look at multiplying to develop the process of factoring trinomials. The value of “a” is 2 and. A polynomial equation or algebraic equation is nothing but an expression consisting of variables and coefficients which only employs the operations of addition, subtraction, multiplication, and non-negative integer exponents. 338… # 5-17 (use calc for #11) 7 more solving by factoring 5. s C JA ilulv VrgiPgMhft 0sw or AeHsEe4rxvueId 6. This video will explain how to factor a polynomial using the greatest common factor, trinomials and special factoring rules. Directions. Generate an unlimited number of custom math worksheets instantly. To simplify this expression, you remove the parentheses by multiplying 5x by each of the three terms inside …. Make a scale drawing of your classroom. Find the scale factor of the dilation. 1) 9 x2 − 1 2) 4n2 − 49 3) 36k2 − 1 4) p2 − 36 5) 2x2 − 18 6) 196n2 − 144. When students have that down, I move on to factoring trinomials. In the multiplication problem , 5 and 4 are factors and 20 is the product. a whole number greater than 1 that. The company has its modern part of hunting and fishing culture, which is always existed there and the catches are made with deep respect in. ) 𝟏 − 𝟐 (𝟐 + 𝟑𝒙). Continue by saying, “Can you name some fruits that would be included in this group?” Allow the children time to answer then say, “Very good!” It’s Fun to Eat Fruits and Veggies! 119 IT’S FUN TO EAT FRUITS AND VEGGIES. Instructions. Similar figures are figures that have the same shape. Equation of Circle. At the end of the day, no matter what a company makes or manages or perfects or distributes, most are in the business of making money. Factoring/ Expanding Quadratics: Worksheet Generator. These worksheets are especially meant for pre-algebra and algebra 1 courses (grades 6-9). Engineering Connection Suppose an engineer draws out a design for a new microscope. The comprehensive lesson plans outlined below provide a detailed list of the Time4Learning third grade math. Celtics co-owner Wyc Grousbeck has donated $1 million toward local efforts to develop a vaccine for COVID-19, according to The Boston Globe. A couple days in we go through this factoring Prezi as a way for students to self-assess and for me to see who knows (and doesn't yet know) how to factor when a is 1. Included are games and activities that teaches how to solve algebraic equations and expressions, Words to Algebra, equation of line, slope and intercept, algebra quizzes and worksheets, examples and step by step solutions. Use conversion factors from the SI system to do the following conversions: a. Monohybrid Crosses. There are a few shortcuts that we can take when multiplying polynomials. Sample answer: domain: all real numbers;. This math worksheet is printable and displays a full page math sheet with Horizontal Multiplication questions. Measure the length and width of the room and anything else you think you need. com Visit http://www. Factoring polynomials can be easy if you understand a few simple steps. The worksheets below give students practice in creating factor trees. 1) (2 m − 2)(4m − 8) 8m2 − 24 m + 16 2) (4r + 4)(r − 4) 4r2 − 12 r − 16 3) (7x + 1)(8x + 8) 56 x2 + 64 x + 8 4) (n + 6)(7n + 4) 7n2 + 46 n + 24 5) (3b + 3)(5b + 8) 15 b2 + 39 b + 24 6) (6v + 8)(2v + 3) 12 v2 + 34 v + 24 7) (5x + 7)(6x − 5) 30 x2 + 17 x − 35 8) (3n − 6)(8n − 6. TIP: Before you can apply the general steps below, make sure to first take out common factors among the coefficients of the trinomial. Sample Problem 2: a. Factoring Polynomials: Classwork/Practice Packet Lesson 1: Using the Greatest Common Factor and the Distributive Property to Factor Polynomials pg. Each of these companies belonged to different industries, viz. Multiplying, we get the original and can see that the terms within the parentheses have no other common factor, so we know the solution is correct. 5 Factoring Polynomials of the form C Factor X2 2 x +1—6 Name Block 2 12 x —x— 10. Showing top 8 worksheets in the category - Scale Drawing. If the larger number is negative, the answer is negative. Every Shakespeare Play Summed Up in a Quote from The Office Dec 27, 2019. “So I can. Continue by saying, “Can you name some fruits that would be included in this group?” Allow the children time to answer then say, “Very good!” It’s Fun to Eat Fruits and Veggies! 119 IT’S FUN TO EAT FRUITS AND VEGGIES. Dimensional Analysis (also called Factor-Label Method or the Unit Factor Method) is a problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value. mini-quiz b zero product property and writing equations when given the roots (zeros) 5. Possible answer: The purpose of this activity was to introduce, define, and practice using conversion factors to find equivalent forms of the same quantity 15. 21 Factor polynomials using a variety of methods including the factor theorem, synthetic division, long division, sums and differences of cubes, and grouping. If we reverse the problem, , we say we have factored 20 into. Sheets Used in Video (Answer Key) Practice for Video Problems #1 Through #6 Practice for Video Problems #1 Through #6 (Answer Key) Solving Algebraic Eq. This game has a single-player feature, and a multi-player option. 4 A wAfl Gl0 Krai ogohOtns7 cr7e rs 4e4rqv3eId A. Questionnaires can be thought of as a kind of written interview. Directions. Factoring Practice I. Com provides free math worksheets for teachers, parents, students, and home schoolers. We hope to add more problem types later. Have Fun Teaching Resources such as worksheets, activities, workbooks, songs, and videos are perfect for teaching. Fun Math Jokeheetsheet Did You Hear About Answers Virtual Filing from Did You Hear About Math Worksheet Answer Key , source: gigidiaries. r2+112v+210] [2] [7x — 5] 35] 2x+10 [7x+21] [3] 18 [2x2 + 8x+ 6] [12r+ 12] [12x+ 9] 36x2+ 144x+ 108 [3] 12x [24x2 + 72x]. There's no better way to find math help online than with Cymath, so also make sure you download our mobile app for iOS and Android today! Learn more than what the answer is - with the math helper. making practice fun algebra 44 answer key. a whole number greater than 1 that has more than two positive factors B. Microsoft Word - WS Factoring Trinomials. And at the end they have a snazzy picture! 1/10/18 - A new Clue Activity for Solving Quadratics by Factoring. To begin this lesson, it is important for you to understand the process of multiplying binomials using the FOIL method. Factor the following using a Difference or Sum of Two Cubes. Scale Factors of Similar Figures Page 1 of 3. com was launched in 2005 with around 400 math worksheets. (The latter can engulf and destroy bacteria, viruses, and other foreign substances. Distribute Algebra tiles and copies of the Factoring Polynomials Using Algebra Tiles activity sheet. factoring numbers negative numbers pi prime numbers square roots Probability Puzzles Ratio/Proportion Statistics Word Problems: Browse Middle School Word Problems Stars indicate particularly interesting answers or good places to begin browsing. Factor each expression. Each student factored all 13 of the problems and chose a color th. Remember, each row, column, and 3x3 square should show the numbers 1 through 9 without repetition. to its conquest by Alexander the Great in 332 B. second method for solving the equations: factoring. Play against the computer in this classic game. Step 2: Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together. m Worksheet by Kuta Software LLC. Factoring polynomials with 4 terms by grouping worksheet - Solution. The player with the greater total when the game ends is the winner. The worksheet is great but the answer key is fantastic! This practice test focuses on quadratics and polynomials. Common Core State Standards CCSS. A comprehensive database of more than 34 polynomial quizzes online, test your knowledge with polynomial quiz questions. ©7 42e0 61n2U UKXu0tga k zSPo0f NtPwCalroe 6 RLhL 4C w. 1) log (u2 v) 3 2) log 6 (u4v4) 3) log 5 3 8 ⋅ 7 ⋅ 11 4) log 4 (u6v5) 5) log 3 (x4 y) 3 Condense each expression to a single logarithm. The Factor Theorem. Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring Lesson 3: Greatest Common Factor and Prime Factors The greatest common factor (GCF) is the largest factor shared by two or more terms. The bus factor [4] of a project refers to the number of team members that can be hit by a bus (or gets sick, leaves the company, goes on. January 14, 2018. 5 x 11 inch paper if that works with your scale factor OR use rolled white chart paper if your scale factor is too big for regular paper. For instance, 6 is a factor of 12, 6, and 18, and x is a factor of each term. All numbers are made up of prime factors, and the list of primes whose product is a particular number are called that number's prime factorization. You can see this because (3x^3 + 6x^2) = (3x_x^2 + 3_2x^2). 1 is a factor of every number. For example, you can factor x3 + x2 - x - 1 by using grouping. 5x + 35 16. More factoring games and puzzles, courtesy of Mr. Every Shakespeare Play Summed Up in a Quote from The Office Dec 27, 2019. Riddles and answers for your enjoyment. More Math Games to Play. 1 is a factor of every number. KEY CONCEPTS AND VOCABULARY Direct Variation- a linear function defined by an equation of the form y=kx, where k ≠ 0. Silent K Worksheets. We provide full solutions with steps for all practice problems. 2x 28x 7 2 4. Factoring is to write an expression as a product of factors. After multiplying the fractions and typing in answers for all 10 problems, check your answers. com is an online resource used every day by thousands of teachers, students and parents. Entrepreneurship creates a new field of business, despite risks and uncertainty. More about privacy & security. *Answer key included* Directions: Print each of the following pages on colored p. For our example above with 12 the complete factorization is, $12 = \left( 2 \right)\left( 2 \right)\left( 3 \right)$ Factoring polynomials is done in pretty much the same manner. Used by more than 50% of US schools. Making statements based on opinion; back them up with references or personal experience. I tend to spend an extra day teaching factoring by grouping. 3 I HM0a xd IeW 3wLi1txh I dIjn zfmiRn1ixt7e o MAnl Tg xekb fr1a e j1 6. 1 Use Properties of Exponents Lesson 5. All for the middle levels of Grade 6, Grade 7, and Grade 8. 7 feet the answer is found by substituting 9 for each x in the equation 5. The key features that make the Ironclaw worth a look can also be found on more expensive mice, but at$59. The first page has the questions; the second page displays the answers. LCM forms an important part of the math curriculum in the United States. One of the largest available anywhere. composite number 3. In the third column, they should write a brief description of the function of the. Then, I show them the two key words (factor, multiply). Similar figures are figures that have the same shape. You should try this method first when faced with a polynomial with four or more terms. Several health conditions, your lifestyle, and your age and family history can increase your risk for heart disease. Spoons-Algebra Style ! The Object Be the first player to hold a set of four cards with the same solution. Factor trees may be used to find the GCF of difficult numbers. Section 9‐2: Factoring by GCF Notes - Part A Example 1: Greatest common factor. Assessment tests aren't graded--they're more like a personality test such as Myers-Briggs. Number puzzles have specific set of rules, you first has to figure out the pattern being followed and then answer the puzzle according to the pattern. Picture dictionary. 1) x2 − 9x + 18 = 0 2) x2 + 5x + 4 = 0 3) n2 − 64 = 0 4) b2 + 5b = 0 5) 35n2 + 22n + 3 = 0 6) 15b2 + 4b − 4 = 0 7) 7p2 − 38p − 24 = 0 8) 3x2 + 14x − 49 = 0 9) 3k2 − 18k − 21 = 0 10) 6k2 − 42k + 72 = 0 11) x2 = 11x − 28 12) k2 + 15k = −56. Print or download free pdf printable worksheet and teach students about Greatest Common Factor. The Quadratic Equation Worksheet Maker will generate a printable worksheet of problems and an answer key. FACTORING TRINOMIALS OF THE FORM x2 + bx + c Factoring with three terms, or trinomials, is the most important technique, especially in further algebra. This answer should be in your own words. We will open a new window containing your custom quadratic equations worksheet. Numerology could also identify exactly when challenges as well as opportunities will occur throughout a person’s life. Interesting Fun Facts About Numerology Here is some quick list of interesting fun facts about numerology: A person’s strengths and weaknesses of character as well as his/her life purpose could be determined using numerology. 3 x y − 24 x 2 − 7 y + 56 x = ( 3 x y − 24 x 2) + ( − 7 y + 56 x) Now check each group for any common factors. Factoring #1: Factoring out the GCF ID: 1 Name_____ Period____ ©3 m2p0z1a1 T dKnurt ha Z 6S Komf5tfwqafrCe4 fL7L1C B. The personality model is analysed by 5 main areas – Neuroticism, Agreeableness, Extroversion, Conscientiousness and Openness. Made in cooperati. The math quizzes can be used for both practice and evaluation. 8A Factoring Methods 8-1 Factors and Greatest Common Factors Lab Model Factoring 8-2 Factoring by GCF Lab Model Factorization of Trinomials 8-3 Factoring x 2 + bx + c 8-4 Factoring ax 2 + bx + c Lab Use a Graph to Factor Polynomials 8B Applying Factoring Methods 8-5 Factoring Special. 2x 28x 7 2 4. Intermediate Algebra Skill Factoring the Difference of Squares Factor each completely. Step 2: Every difference of squares problem can be factored as follows: a 2 – b 2 = (a + b)(a – b) or (a – b)(a + b). 2 Factoring the Difference of Two Squares 5. All for the middle levels of Grade 6, Grade 7, and Grade 8. H Worksheet. Our grade 6 factoring worksheets complement our online math program. making practice fun algebra 44 answer key. This game has a single-player feature, and a multi-player option. Factoring polynomials and operations worksheets pdf. Dividing Polynomials: Polynomial by a Quadratic Dividing Polynomials: Mix Describe the Left and Right Behavior of the Graph Graph. Smoking is a key risk factor for heart disease. The website was created in September 2005. SUMMARY OF ALGEBRA I MATHEMATICS CONTENT Edgenuity Algebra I strictly adheres to the content specified by the Common Core State Standards in conjunction with Louisiana Student Standards for Mathematics. Factor the common factor out of each expression. The generator includes only very simple problems with linear expressions. That is the reason for factoring things in this way. This can be factored into (x+2)(x+3)=0. Or click the "Show Answers" button at the bottom of the page to see all the answers at once. 8 8 = x x b. Tumour necrosis factor (TNF), a naturally occurring protein that is produced in the human body by the phagocytic cells known as macrophages. LCM forms an important part of the math curriculum in the United States. Correct Answer Your Answer; 1: x 2 + 3 x + 2: Solution The factored form of the trinomial will take the form of (x + ?)(x + ?) Front coefficient is 1. This is not possible with all polynomials, but it's a good approach to check first. If so, factor out the GCF. Each of these companies belonged to different industries, viz. 27 T 7+ 8 U : c. I accept both answers from students, but ask them to think about why these are factors. Use the settings and options below to generate the type of questions you want to include on the worksheet. It is visited by 14,000 - 28,000 people daily, mostly from USA, Bulgaria, Russia, India, Philippines, Ukraine, Serbia. Step #2: Split the middle term bx = n1x + n2x. Silent K Worksheets. The Fun Music Company gives you instant access to the music resources you require within minutes, so you can have something for your class the next day! We also deliver high quality programs on CD-ROM, so that you can prepare the lessons at home, and also take them to school and use as needed. Filter by Grade, Subject, and Skill. These multiplying polynomials worksheets with answer keys encompass polynomials to be multiplied by monomials, binomials, trinomials and polynomials; involving single and multivariables. Math-Drills. 1 is a factor of every number. Student Workbook Answer Key: PRE-LESSON NESPA Lesson One AK 2 EG-2007-01-203-ARC Name: Date: Pre-Lesson Activity Step 1: On the back of this paper draw a picture of our solar system. I was worried with this activity that students might just look at the answer bank and not try the factoring on their own, but this was not an issue at all. D, who wrote the. Algebra Tiles. Depending on the age of the student, most teachers allow students from 3 to 5 minutes to complete the 32 problems on each quiz. Math workbook 1 is a content-rich downloadable zip file with 100 Math printable exercises and 100 pages of answer sheets attached to each exercise. Take the polynomial and separate it into two groups. The flow chart below uses three variables of which you can choose the starting value for M. Sample answer: domain: all real numbers;. Funny turkey jokes are good any time of the year, but especially around holidays like Thanksgiving and Christmas. tan2 x tan2 x+1 = sin2 x Solution: LHS = tan2 x tan2 x+1 = sinx cosx 2 sinx cosx 2 +1 = sin2 x cos2 x sin2 x cos2 x +1 = sin2 x cos2 x. These math worksheets cover factoring to prime numbers as well as determining the greatest common factor (GCF) or Least Common Multiple (LCM) of two numbers. Today we are going to practice factoring & solving (or finding zeros) of higher degree polynomials. One of the numbers I wrote down must be 0 in order for them to. Select one of the links below to get started. Learn how to factor quadratics that have the "difference of squares" form. Answer Key: Yes. Factor each of the following polynomials A) x y yz z2 2 2− + −2 B) 3 243x4 − C) 2 16x3 −. 36 36 = x x x a. Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring Lesson 3: Greatest Common Factor and Prime Factors The greatest common factor (GCF) is the largest factor shared by two or more terms. A trinomial is a 3 term polynomial. 1-7 The Distributive Property 7-1 Zero and Negative Exponents 8-2 Multiplying and Factoring 10-2 Simplifying Radicals 11-3 Dividing Polynomials 12-7 Theoretical and Experimental Probability Absolute Value Equations and Inequalities Algebra 1 Games Algebra 1 Worksheets algebra review solving equations maze answers Cinco De Mayo Math Activity. Abiotic factor definition, a nonliving condition or thing, as climate or habitat, that influences or affects an ecosystem and the organisms in it: Abiotic factors can determine which species of organisms will survive in a given environment. Step #3: x(x+2) + 4(x + 2) (x + 2) (x + 4) Exp. Guidelines 1. 5 liters to cubic decimeters. The problems may be selected from two different degrees of difficulty. Parents can expect to see subjects covered including identifying transformations and symmetry, demonstrating fractions, problem solving and more. We hope that you find exactly what you need for your home or classroom!. All numbers are made up of prime factors, and the list of primes whose product is a particular number are called that number's prime factorization. 2x 28x 7 2 4. Step #2: Split the middle term bx = n1x + n2x. x3 4x2 5x 2 B. The Factor Trees Gizmo has two modes. Some of the worksheets displayed are daffynition decoder answer key angles factoring quadratic expressions daffynition decoder answer key angles daffynition decoder answers sleeping bag pdf daffynition decoder answer key basic geometric terms daffynition decoder answer key ebook daffynition decoder answers europe and underground garage. *Answer key included* Directions: Print each of the following pages on colored p. Preschool Math Games Printable. In this lesson we will study polynomials that can be factored using the Greatest Common Factor. Difficulty Popularity. Notice that this reveals yet another common factor between the two terms: x + 4 \goldD{x+4} x + 4 start color #e07d10, x, plus, 4, end color #e07d10. 1 – 3 Stress experiences can be emotionally or physiologically challenging and activate stress responses and adaptive processes to regain homeostasis. 338… # 5-17 (use calc for #11) 7 more solving by factoring 5. The first page has the questions; the second page displays the answers. Over 300 Algebra worksheets and growing! Printable in convenient PDF format. Example: Multiply the term using distribution. a) 12 and 18. Fifth Grade Math Curriculum: What Students Will Learn. In division, divisor and quotient are both factors of the dividend if the remainder is zero. Using Arrays to Show Multiplication Concepts: Overview Students can more readily develop an understanding of multiplication concepts if they see visual representations of the computation process. In your drawing, show the different sizes of the planets and where they are located. cell cycle and mitosis worksheet answers Review and Specs. FOIL and Factoring Trinomials Find each product. The correct answer is m = 8. Enjoy these free pintable sheets. 9x3y Factoring Polynomials: Factoring Polynomials Completely (Review) in the 1ox with the exercise riuml,er. Factoring Polynomials Solutions Key 223 Holt McDougal Algebra 1 7-1 7 CHAPTER CS10_A1_MESK710372_C07. Working on a team increases the bus factor for a project. Parents can expect to see subjects covered including identifying transformations and symmetry, demonstrating fractions, problem solving and more. August 22, 2018. “We’re downwind,” he says. striped squash (S) is dominant to spotted squash (s) If a. Worksheet about finding the greatest common factor of two numbers. Solutions to equations and inequalities are defined in terms. Factoring #1: Factoring out the GCF ID: 1 Name_____ Period____ ©3 m2p0z1a1 T dKnurt ha Z 6S Komf5tfwqafrCe4 fL7L1C B. Copy this to my account. Multiplying both sides of the equation by the common denominator gives , so. Consider the polynomial $$80v^2u - 8v^3 + 40v^2$$ We wish to find the greatest common factor of the three different terms above. Over 300 Algebra worksheets and growing! Printable in convenient PDF format. Is (x + 1) a factor of f(x) = x 3 + 2x 2 − 5x − 6? Answer. Answers will vary. I can solve by factoring. Mountain Ranges. , Cement, Information Technology (IT), Auto, and Steel, in no particular order. factoring numbers negative numbers pi prime numbers square roots Probability Puzzles Ratio/Proportion Statistics Word Problems: Browse Middle School Word Problems Stars indicate particularly interesting answers or good places to begin browsing. Problem 1 : An art supply store sells several sizes of drawing triangles. Coloring is all the rage and I decided to create a project for my students that allowed them to show their creative side! I started by finding this outline of a mandala online. Greatest Common Factor - Level 3. All are dilations of a single basic triangle. WorksheetWorks. Eat factors of the given multiple. Outline everything in black. The problems may be selected from two different degrees of difficulty. All Factors of a Number - This tool will find all the factors of a number (not just the prime factors); Common Factors Calculator - Enter two integers, then click the button. After having gone through the stuff given above, we hope that the students would have understood "Factoring polynomials with 4 terms by grouping worksheet ". 6 Word Problems Key. 4: group 5: factor each parenthesis. The key is to understand what level of detail you need to achieve the aims of your project so that you can incorporate the right blend of quantitative and qualitative questions. Then, find your answer on the corresponding ornament and color! 1. 5x3 29 x2 19 x 5 D. Note: You will find more Equations & Expressions worksheets here and more worksheet generators here. com Visit http://www. Dimensional analysis (also known as the factor-label method or unit-factor method) is by far the most useful math trick you'll ever learn. Sometimes it is impossible to factor a polynomial by finding the greatest common factor. Grade: 6th to 8th, High School. Because both of these techniques manipulate and rewrite terms, sometimes they can produce solutions that don't work in the original form of the equation. Generate an unlimited number of custom math worksheets instantly. j b yA ol dl r XrBiEgoh 5t7s a RrmePs3ecr4v8e qd g. Showing top 8 worksheets in the category - Pizzazz Answers. Pure water at 25° C has a density of 1. If you can't factor it write "prime" Factor the common factor out of each expression. 7 Fictional Characters Whose Names Give Away the Story Jan 10, 2020. 125x3 64 15. Question: Discuss about the Case Of Organizational Behavior. ANSWER KEY The face ends up making a clown with the following features/colors. When you're done, be sure to check out the unique spiral and bullseye multiplication worksheets to get a. What is the largest possible answer that she could have gotten?. midpoint and distance formula worksheet. Monohybrid Crosses. Elementary Algebra Skill Solving Quadratic Equations by Factoring Solve each equation by factoring. 1) x3 + 8 2) a3 + 64 3) a3 + 216 4) 27 + 8x3 5) a3 − 216 6) 64x3 − 27 7) 27m3 − 125 8) x3 − 64. ) Convert your distance from school to home from miles to inches. Checking Your Answers. Do you have a personal observation which may help others? Free Math Help - Submit your questions, comments, and suggestions using. History Answers to Kids Quiz Two. There is one special case for factoring that you may or may not need, depending upon how your book is structured and how your instructor intends to teach factoring quadratics. About seventeen drinks. membrane structure and function worksheet Concept. Dimensional Analysis Worksheet Use Dimensional Analysis to solve the following problems. Basic plans are free for teachers and students, and plans for your whole school or. Punnett Squares Answer Key. Preview Print Answers. True! Disney films such as Snow White and the Seven Dwarfs and Tangled stem from. Avoid numbers that are not factors of the given multiple. 4 610 prenumeranter. Math 154 :: Elementary Algebra Chapter 7 — Answers Section 7. forcats appeared on CRAN in August 2016. Answer, Find, and Shade Greatest Common Factor - Level 3. One of the numbers I wrote down must be 0 in order for them to. Students may come up with either 10 and 1, 5 and 2 or 15 and 1, 5 and 3 respectively. 1) -12b - 16 2) -4n2 + 4n 3) -4x3 - 16x 4) n2 + n 5) 2b3 - 3b2 6) 5v6 - 15v5 7) r4 + r2 8) -12v - 4v5. This pack contains 10 half page QuickQuizzes. Then determine which number is a perfect number. 2 Factoring the Difference of Two Squares 5. · Fraction-Decimal Conversion. Factoring is also the opposite of Expanding: In the previous example we saw that 2y and 6 had a common factor of 2. Solve each equation by factoring. Remember, each row, column, and 3x3 square should show the numbers 1 through 9 without repetition. We’ve all encountered people like Fred in our lives. Plus model problems explained step by step. so the scale factor from A to B is 2:5. 27 T 7+ 8 U : c. Our grade 6 factoring worksheets complement our online math program. Used by more than 50% of US schools. Gives correct answers: Number of triangles Long beads Round beads 1 3 3 2 6 5 3 9 7 4 12 9 Partial credit 3 or 4 correct answers 2 correct answers 3 (2) (1) 3 3. Do you have a personal observation which may help others? Free Math Help - Submit your questions, comments, and suggestions using. Write down all the numbers which have 6 & 7 as a factor: _____ 15. Topic Jumbled Words. Factoring vs. Help your child get ahead with Education resources, designed specifically with parents in mind. For example, 4 is the greatest common factor of the two numbers 4 and 20. A couple days in we go through this factoring Prezi as a way for students to self-assess and for me to see who knows (and doesn't yet know) how to factor when a is 1. This site is free for the users because of the. Explore all of our factoring worksheets, from factoring numbers under 50 to prime factor trees, GCFs and LCMs. In Build mode, you can build numbers by multiplying primes together. Simplifying an expression often means removing a pair of parentheses; factoring an expression often means applying them.