# Remainder Theorem

## Polynomials and the Remainder Theorem

If your teacher presented you with the following problem, what would you think? Would you know how to solve it? It certainly does not look like something you would see in a math class. It looks more like a secret code. However, it is the remainder theorem formula.

This remainder theorem formula is used with polynomials. By definition, a polynomial is an expression that contains coefficients and variables.

Simply put, it is an expression with letters and numbers in it. Sometimes, there are more letters than numbers, but the letters represent numbers.

Now, keep in mind that a theorem, as in the remainder theorem, is a proof. This means you are trying to prove the theory and not necessarily get a quick numerical answer.

Before we get into the remainder theorem, let’s find out what exactly a polynomial is. Look at the polynomial below. As you can see in the example below, there are variables (letters) and coefficients (numbers) as well as exponents.

The coefficients are the number (or they could be letters) that are shown first before the variable. The variable is almost always a letter that represents a number.

Now, one other thing you will need to know about the remainder theorem is that a linear factor is involved. So, what is a linear factor? In layman’s terms, a linear factor is just a simple equation like the one shown below. It is also called a degree of 1.

To recap, you have learned about the two parts of a remainder theorem, the polynomial, and the linear factor. Now, it is time to dive into the formula.

## Remainder Theorem Formula

Let’s look again at the remainder theorem formula. In this formula, a polynomial is divided by a linear factor. The “p(x)” represents a polynomial with the variable of an unknown amount. With the remainder theorem, you are trying to find the remainder from a division problem.

Let’s review a division problem to understand the terms first before we go any deeper into this polynomial division problem.

FACT: Think of working a polynomial remainder theorem problem like a long division problem!

The polynomial division problem has similar terms. The first part, p(x), means that this is a polynomial with an unknown x-variable. The linear factor is (x-c), and the “c” means it is just an unknown value, and it could be any number.

The “q” is the quotient, and “r” is the remainder. The “x” is again an unknown number.

EXAMPLE
Now, it is time to work on an example using this formula. Let’s look at an example. The polynomial is below.

$$p(x)\;=\;2^{2x}\;+\;3x\;+1$$

The linear factor is $$x\;+2.$$

So, we will divide $$2x^2\;+3x\;+1\;$$ by $$\;x+2.$$

To set it up, you place the polynomial as the dividend, and the linear factor is the divisor. So,  $$2x$$ times $$x$$ is equal to $$2x^2$$and $$2x$$ times 2 is equal to $$4x.$$

Now, you subtract. $$2x^2$$ minus $$2x^2$$ is zero. Then you subtract
$$3x$$ and $$4x$$. However, you are subtracting a positive, so this means you subtract. $$3x$$minus $$4x$$is equal to negative $$(-)x.$$

Then you bring down the plus sign and the one. So, you have negative $$(-) x$$ plus 1.
Negative $$(-1)$$ times $$x+2$$ equals negative $$(-) x$$ minus 2. When you subtract negative $$x$$ minus 2 from negative $$x$$ plus 1, you need to realize that you have two negatives that turn into a positive.

Therefore, you have negative $$x$$ plus 1 plus $$x$$ plus two. The negative x plus positive x cancels out. The remainder is 3.

The final answer would be $$2x-1$$ with a remainder of 3.

## Why do we use the remainder theorem?

Like many people, you may wonder where in the world you are going to use this. First, if you are taking an Algebra or other math class, then obviously you will use it there. For the most part, this is not a skill that you will be using every day.

Instead, you will most likely be using it once in a great, great, while. However, when you are trying to find the length of a flat surface, you may be using this theorem.

The surface area will represent the polynomial and the width will represent the linear factor. When you divide those two numbers, then you will get the answer with the remainder just like a polynomial in the remainder theorem.

Let’s look at a word problem using this theorem.

FACT: Remember when you subtract in a long division problem, it might change the sign on the second number. If you look at the problem above, it shows where the subtracting changed to adding.