Factoring and the Distributive Property

Learning Objective

Introduction

FactorsFor any number `x`, the numbers that can be evenly divided into `x` are called factors of `x`. For example, the number `20` has the factors `1`, `2`, `4`, `5`, `10`, and `20`. are numbers that multiply together to produce another number. For example, `2` and `10` are factors of `20`, as are `4` and `5` and `1` and `20`FactoringThe process of breaking a number down into its multiplicative factors. Every number `x` has at least the numbers `1` and `x` as factors. is the process of breaking a number down into its multiplicative factors. Factoring has many applications in mathematics; breaking a value down into smaller pieces gives us new ways to understand and manipulate it.

Just as any integer can be written as the product of factors, so too can any monomialA number, a variable, or a product of a number and one or more variables with whole number exponents, such as `-5`, `x`, and `8xy^3`. or polynomialA monomial or sum of monomials, like `4x^2 + 3x - 10`. be expressed as a product of factors. Factoring is very helpful in simplifying and solving equations using polynomials.

The factors of a value always include what are called prime factorsA factor that has no factors but `1` and itself. For example, `2` is a prime factor of `12` because its only factors are `1` and `2`, while `6` is not a prime factor of `12` because it has more factors than `1` and `6` (i.e. `2` and `3`)., which have no factors but `1` and themselves (in other words, they are prime numbers). For example, `2` is a prime factor of `20` because its only factors are `1` and `2`. But `10` is not a prime factor because `10` can be factored into not just `1` and `10` but also `2` and `5`.

The process of breaking a number down into its prime factors is called prime factorizationThe process of breaking a number down into its prime factors..

Factoring Numbers

Let’s use a common factoring task—finding the greatest common factor (GCF)The largest factor that two numbers have in common. of two whole numbers—as a jumping off point. The GCF of two numbers is the largest number that is a factor of both of the numbers. To find the GCF, factor both numbers to find their prime factors, identify the prime factors they have in common, and then multiply those together.

Example

Problem

Find the greatest common factor of `210` and `168`.

 

`210 = bb2*bb3*5*bb7`

 

`168 = bb2 * 2 * 2 * bb3 * bb7`

 

`"GCF" = bb2 * bb3 * bb7`

Answer

`"GCF" = 42`

Because the GCF is the product of the prime factors that these numbers have in common, we know that it is a factor of both numbers. (If you want to test this, go ahead and divide both `210` and `168` by `42`—they are both evenly divisible by this number!)

It is also worth noting here that while both `210` and `168` are broken down into their prime factors in the example above, we could have gotten the same answer using any old factors. We could have factored the numbers as shown below and still arrived at the same GCF of `42`. But this method is a little tricky, because we have to be able to recognize that the non-common factors (in this case `5` and `4`) don’t share any common factors other than `1`.

Example

Problem

Find the greatest common factor of `210` and `168`.

 

`210 = bb6 * 5 * bb7`

 

`168 = 4 * bb6 * bb7`

 

`"GCF" = bb6 *bb 7`

Answer

`"GCF" = 42`

Factoring Monomials

Finding the greatest common factor in a set of monomials is not very different from finding the GCF of two whole numbers. Our method remains the same: we will factor each monomial independently, look for common factors, and then multiply them to get the GCF.

Let’s begin by finding the greatest common factor of `15b^2` and `20b`. This problem certainly looks a bit different than ones we are used to; instead of having just whole numbers, this one has the variable terms `b^2` and `b` present as well. Let’s factor these monomials first, remembering that the variable terms should be factored as well. Can you spot any common factors between `15b^2` and `20b`?

Example

Problem

Find the greatest common factor of `15b^2` and `20b`.

 

`15b^2 = 3 * bb5 * bb b * b` 

 

`20b = 4 * bb5 * bb b` 

 

`"GCF" = bb5* bb b` 

Answer

`"GCF" = 5b`

The monomials have the factors `5` and `b` in common, which means their greatest common factor is `5 * b`, or simply `5b`.

Let’s try one more problem before moving on: finding the greatest common factor of `81c^4d` and `45c^2d`. We will first factor `81c^4d` and `45c^2d`, then look for common factors. Notice that these monomials each have two variables.

Example

Problem

Find the greatest common factor of `81c^4d` and `45c^2d`

 

`81c^4d = bb3 * bb3 * 3 * 3 * bbc * bbc* c * c * bb d` 

 

`45c^2d = bb3 * bb3 * 5 * bb c * bb c * bb d` 

 

`"GCF" = bb3 * bb3 * bb c * bb c * bb d` 

Answer

`"GCF" = 9c^2d`

This example presented `81c^4d` and `45c^2d` as products of prime factors. As you become more familiar with factoring monomials, you will begin to see common factors without needing to reduce monomials to their prime factors. For example, you may have factored the monomials like this:

Example

Problem

Find the greatest common factor of `81c^4d` and `45c^2d`.

 

`81c^4d = bb9 * 9* bb c^bb2* c^2 * bb d` 

 

`45c^2d = bb9 * 5* bbc^bb2 * bb d` 

 

`"GCF" = bb9 * bbc^bb2 * bb d` 

Answer

`"GCF" = 9c^2d`

Notice that this factoring still leads to the same GCF: `9c^2d`.

Find the greatest common factor of `56xy` and `16y^3`.

 

A) `8`

 

B) `8y`

 

C) `16y`

 

D) `8xy^3`

 

Factoring Polynomials

When two or more polynomials are combined (either added or subtracted), the resulting expression is called a polynomial. We can factor polynomials using some of the same logic that we applied to factoring monomials.

Remember that factoring is the process of breaking a number down into its multiplicative components. As you look at the examples of simple polynomials below, try to identify factors that the terms of the polynomial have in common.

Polynomial

Monomials

Common Factors

`6x + 9`

`6x` and `9`

`3` is a factor of `6x` and `9`

`a^2 - 2a`

`a^2` and `- 2a`

`a` is a factor of `a^2` and `-2a`

`4c^3 + 4c`

`4c^3` and `4c`

`4` and `c` are factors of `4c^3` and `4c`

To factor a polynomial, first identify the greatest common factor of the monomial terms. Then use the distributive propertyStates that the product of a number and a sum equals the sum of the individual products of the number and the addends: for all real numbers `a`, `b`, and `c`, `a(b + c) = ab + ac`. to rewrite the polynomial as the product of the GCF and the other parts of the polynomial. (Remember, the distributive property says that `a(b + c) = a*b + a * c`. This rewritten expression is the factored form of a polynomialA polynomial written as a product of factors, and each non-monomial factor has no common factors in its terms..

Polynomial

GCF

Factored Form

`6x + 9`

`3`

`3(2x + 3)`

`a^2 - 2a`

`a`

`a(a - 2)`

`4c^3 + 4c`

`4c`

`4c(c^2 + 1)`

Look at how we can “pull out” the common factor in each polynomial. We know that both terms in a polynomial are divisible by their GCF, so we can rewrite each polynomial as a product of the GCF and the combined "left over" factors of each monomial.

Let's go through the process of factoring a polynomial, step by step.

Example

Problem

Factor `24d^2 - 18d`

 

`bb6 * 4 * bbd*d`  

 

Factor `24d^2`.

 

`-1 * bb6 * 3*bbd` 

 

Factor `-18d`.

 

`bb6*bbd` 

`6d`

 

Find the GCF.

 

 

`24d^2 = 6d * 4d`

`-18d = 6d * -3`

 

Rewrite each monomial with the GCF as one factor.

 

 

`6d(4d) - 6d(3)`

 

Rewrite the polynomial expression using the factored monomials in place of the original terms.

 

`6d(4d - 3)`

 

Use the distributive property to pull out the GCF.

Answer

`6d(4d - 3)`

 

 

           

To reassure ourselves that this is correct, we can multiply `6d * (4d - 3)`, checking to see if we get the original form of the polynomial, `24d^2 - 18d`

The factored form of a polynomial is one in which the polynomial is written as a product of factors, and each non-monomial factor has no common factors in its terms. For example, the factors of `6d(4d - 3)` are `6``d`, and `4d - 3`. The two terms of `4d - 3` have no common factors.

As a final example, let’s return to a set of monomials for which we found the GCF earlier on: `81c^4d` and `45c^2d`. If we add these two quantities and make a single polynomial, how could we factor it?

Example

Problem

Factor `81c^4d + 45c^2d`

 

`bb9 * 9*bbc^bb2*c^2*bbd` 

Factor `81c^4d`

 

`bb9 * 5*bbc^bb2*bbd` 

Factor `45c^2d`

 

`bb9*bbc^bb2*bbd` 

`9c^2d`

Find the GCF.

 

 

 

`81c^4d = 9c^2d*9c^2`

 

`45c^2d = 9c^2d*5`

Rewrite each monomial as the product of the GCF and the remaining terms.

 

`9c^2d*9c^2 + 9c^2d*5`

Rewrite the polynomial expression using the factored monomials in place of the original terms.

 

`9c^2d*(9c^2 + 5)`

Use the distributive property to pull out the GCF.

Answer

`9c^2d*(9c^2 + 5)`

 

Knowing the greatest common factor of the two monomials helped us factor the polynomial as a whole. Having identified the GCF, we use the distributive property to arrive at the factored form: `9c^2d(9c^2 + 5)`

Factor: `8r^4 - 11r^3`

 

A) `88(r^4 - r^3)`

 

B) `8r(r^3 - 3)`

 

C) `r^3(r - 1)`

 

D) `r^3(8r - 11)`

 

Summary

Any monomial or polynomial can be expressed as a product of factors. We use some of the same techniques that we apply to factoring integers to factor polynomials. To factor a polynomial, identify the greatest common factor of the monomial terms, and then use the distributive property to rewrite the expression. Once a polynomial in `a * b + a * c` form has been rewritten as `a(b + c)`, where `a` is the GCF, we say that the polynomial is in factored form.