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### Course: Algebra (all content) > Unit 10

Lesson 10: Introduction to factorization# Intro to factors & divisibility

Learn what it means for polynomials to be factors of other polynomials or to be divisible by them.

## What we need to know for this lesson

A $x$ , like $3{x}^{2}$ . A $3{x}^{2}+6x-1$ .

**monomial**is an expression that is the product of constants and nonnegative integer powers of**polynomial**is an expression that consists of a sum of monomials, like## What we will learn in this lesson

In this lesson, we will explore the relationship between factors and divisibility in polynomials and also learn how to determine if one polynomial is a factor of another.

## Factors and divisibility in integers

In general, two integers that multiply to obtain a number are considered

**factors**of that number.For example, since $14=2\cdot 7$ , we know that $2$ and $7$ are $14$ .

**factors**ofOne number is

**divisible**by another number if the result of the division is an integer.For example, since $\frac{15}{3}}=5$ and $\frac{15}{5}}=3$ , then $15$ is divisible by $3$ and $5$ . However, since $\frac{9}{4}}=2.25$ , then $9$ is $4$ .

*not divisible*byNotice the mutual relationship between factors and divisibility:

Since ${14}={2}\cdot 7$ (which means $2$ is a factor of $14$ ), we know that $\frac{{14}}{{2}}}=7$ (which means $14$ is divisible by $2$ ).

In the other direction, since $\frac{{15}}{{3}}}=5$ (which means $15$ is divisible by $3$ ), we know that ${15}={3}\cdot 5$ (which means $3$ is a factor of $15$ ).

This is true in general: If $a$ is a factor of $b$ , then $b$ is divisible by $a$ , and vice versa.

## Factors and divisibility in polynomials

This knowledge can be applied to polynomials as well.

When two or more polynomials are multiplied, we call each of these polynomials

**factors**of the product.For example, we know that $2x(x+3)=2{x}^{2}+6x$ .
This means that $2x$ and $x+3$ are factors of $2{x}^{2}+6x$ .

Also, one polynomial is

**divisible**by another polynomial if the quotient is also a polynomial.For example, since $\frac{6{x}^{2}}{3x}}=2x$ and since $\frac{6{x}^{2}}{2x}}=3x$ , then $6{x}^{2}$ is divisible by $3x$ and $2x$ . However, since $\frac{4x}{2{x}^{2}}}={\displaystyle \frac{2}{x}$ , we know that $4x$ is $2{x}^{2}$ .

*not divisible*byWith polynomials, we can note the same relationship between factors and divisibility as with integers.

In general, if $p=q\cdot r$ for polynomials $p$ , $q$ , and $r$ , then we know the following:

and$q$ are factors of$r$ .$p$ is divisible by$p$ and$q$ .$r$

### Check your understanding

## Determining factors and divisibility

### Example 1: Is $24{x}^{4}$ divisible by $8{x}^{3}$ ?

To answer this question, we can find and simplify $\frac{24{x}^{4}}{8{x}^{3}}$ . If the result is a monomial, then $24{x}^{4}$ is divisible by $8{x}^{3}$ . If the result is not a monomial, then $24{x}^{4}$ is not divisible by $8{x}^{3}$ .

Since the result is a monomial, we know that $24{x}^{4}$ is divisible by $8{x}^{3}$ . (This also implies that $8{x}^{3}$ is a factor of $24{x}^{4}$ .)

### Example 2: Is $4{x}^{6}$ a factor of $32{x}^{3}$ ?

If $4{x}^{6}$ is a factor of $32{x}^{3}$ , then $32{x}^{3}$ is divisible by $4{x}^{6}$ . So let's find and simplify $\frac{32{x}^{3}}{4{x}^{6}}$ .

Notice that the term $\frac{8}{{x}^{3}}$ is $4{x}^{6}$ $32{x}^{3}$ .

*not*a monomial since it is a quotient, not a product. Therefore we can conclude that*is not*a factor of### A summary

In general, to determine whether one polynomial $p$ is divisible by another polynomial $q$ , or equivalently whether $q$ is a factor of $p$ , we can find and examine $\frac{p(x)}{q(x)}$ .

If the simplified form is a polynomial, then $p$ is divisible by $q$ and $q$ is a factor of $p$ .

### Check your understanding

## Challenge problems

## Why are we interested in factoring polynomials?

Just as factoring integers turned out to be very useful for a variety of applications, so is polynomial factorization!

Specifically, polynomial factorization is very useful in solving quadratic equations and simplifying rational expressions.

If you'd like to see this, check out the following articles:

## What's next?

The next step in the factoring process involves learning how to factor monomials. You can learn about this in our next article.

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