# 4.4 Combining polynomials using addition and subtraction

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: understand the concept of like terms, be able to combine like terms, be able to simplify expressions containing parentheses.

## Overview

• Like Terms
• Combining Like Terms
• Simplifying Expressions Containing Parentheses

## Like terms

Terms whose variable parts, including the exponents, are identical are called like terms . Like terms is an appropriate name since terms with identical variable parts and different numerical coefficients represent different amounts of the same quantity. As long as we are dealing with quantities of the same type we can combine them using addition and subtraction.

## Simplifying an algebraic expression

An algebraic expression can be simplified by combining like terms.

## Sample set a

Combine the like terms.

$6\text{\hspace{0.17em}}\text{houses}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\text{houses}=\text{\hspace{0.17em}}10\text{\hspace{0.17em}}\text{houses}$ . 6 and 4 of the same type give 10 of that type.

$6\text{\hspace{0.17em}}\text{houses}+4\text{\hspace{0.17em}}\text{houses}+2\text{\hspace{0.17em}}\text{motels}=10\text{\hspace{0.17em}}\text{houses}+2\text{\hspace{0.17em}}\text{motels}$ . 6 and 4 of the same type give 10 of that type. Thus, we have 10 of one type and 2 of another type.

Suppose we let the letter $x$ represent "house." Then, $6x+4x=10x$ . 6 and 4 of the same type give 10 of that type.

Suppose we let $x$ represent "house" and $y$ represent "motel."

$6x+4x+2y=10x+2y$

## Practice set a

Like terms with the same numerical coefficient represent equal amounts of the same quantity.

Like terms with different numerical coefficients represent .

different amounts of the same quantity

## Combining like terms

Since like terms represent amounts of the same quantity, they may be combined, that is, like terms may be added together.

## Sample set b

Simplify each of the following polynomials by combining like terms.

$2x+5x+3x$ .
There are $2x\text{'}\text{s}$ , then 5 more, then 3 more. This makes a total of $10x\text{'}\text{s}$ .

$2x+5x+3x=10x$

$7x+8y-3x$ .
From $7x\text{'}\text{s}$ , we lose $3x\text{'}\text{s}$ . This makes $4x\text{'}\text{s}$ . The $8y\text{'}\text{s}$ represent a quantity different from the $x\text{'}\text{s}$ and therefore will not combine with them.

$7x+8y-3x=4x+8y$

$4{a}^{3}-2{a}^{2}+8{a}^{3}+{a}^{2}-2{a}^{3}$ .
$4{a}^{3},\text{\hspace{0.17em}}8{a}^{3},$ and $-2{a}^{3}$ represent quantities of the same type.

$4{a}^{3}+8{a}^{3}-2{a}^{3}=10{a}^{3}$

$-2{a}^{2}$ and ${a}^{2}$ represent quantities of the same type.

$-2{a}^{2}+{a}^{2}=-{a}^{2}$

Thus,

$4{a}^{3}-2{a}^{2}+8{a}^{3}+{a}^{2}-2{a}^{3}=10{a}^{3}-{a}^{2}$

## Practice set b

Simplify each of the following expressions.

$4y+7y$

$11y$

$3x+6x+11x$

$20x$

$5a+2b+4a-b-7b$

$9a-6b$

$10{x}^{3}-4{x}^{3}+3{x}^{2}-12{x}^{3}+5{x}^{2}+2x+{x}^{3}+8x$

$-5{x}^{3}+8{x}^{2}+10x$

$2{a}^{5}-{a}^{5}+1-4ab-9+9ab-2-3-{a}^{5}$

$5ab-13$

## Simplifying expressions containing parentheses

When parentheses occur in expressions, they must be removed before the expression can be simplified. Parentheses can be removed using the distributive property.

## Sample set c

Simplify each of the following expressions by using the distributive property and combining like terms.

#### Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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LITNING Reply
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LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Rafiq
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Mahi
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Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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Bharti
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Damian Reply
absolutely yes
Daniel
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Smarajit Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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