# 9.7: Factoring Polynomials Completely

## Learning Objectives

- Factor out a common binomial.
- Factor by grouping.
- Factor a quadratic trinomial where .
- Solve real world problems using polynomial equations.

## Introduction

We say that a polynomial is **factored completely** when we can’t factor it any more. Here are some suggestions that you should follow to make sure that you factor completely:

- Factor all common monomials first.
- Identify special products such as difference of squares or the square of a binomial. Factor according to their formulas.
- If there are no special products, factor using the methods we learned in the previous sections.
- Look at each factor and see if any of these can be factored further.

**Example 1**

*Factor the following polynomials completely.*

a)

b)

c)

**Solution**

a) Factor out the common monomial. In this case 6 can be divided from each term:

There are no special products. We factor as a product of two binomials:

The two numbers that multiply to 6 and add to -5 are -2 and -3, so:

If we look at each factor we see that we can factor no more.

The answer is .

b) Factor out common monomials:

We recognize as a difference of squares. We factor it as .

If we look at each factor we see that we can factor no more.

The answer is .

c) Factor out common monomials:

We recognize as a perfect square and factor it as .

If we look at each factor we see that we can factor no more.

The answer is .

**Example 2**

Factor the following polynomials completely:

a)

b)

**Solution**

a) Factor out the common monomial. In this case, factor out -2 rather than 2. (It’s always easier to factor out the negative number so that the highest degree term is positive.)

We recognize expression in parenthesis as a difference of squares. We factor and get:

If we look at each factor we see that the first parenthesis is a difference of squares. We factor and get:

If we look at each factor now we see that we can factor no more.

The answer is .

b) Factor out the common monomial:

We recognize as a perfect square and we factor it as .

We look at each term and recognize that the term in parentheses is a difference of squares.

We factor it and get , which we can rewrite as .

If we look at each factor now we see that we can factor no more.

The final answer is .

## Factor out a Common Binomial

The first step in the factoring process is often factoring out the common monomials from a polynomial. But sometimes polynomials have common terms that are binomials. For example, consider the following expression:

Since the term appears in both terms of the polynomial, we can factor it out. We write that term in front of a set of parentheses containing the terms that are left over:

This expression is now completely factored.

Let’s look at some more examples.

**Example 3**

*Factor out the common binomials.*

a)

b)

**Solution**

a) has a common binomial of .

When we factor out the common binomial we get .

b) has a common binomial of .

When we factor out the common binomial we get .

## Factor by Grouping

Sometimes, we can factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called **factor by grouping.**

The next example illustrates how this process works.

**Example 4**

*Factor* .

**Solution**

There is no factor common to all the terms. However, the first two terms have a common factor of 2 and the last two terms have a common factor of . Factor 2 from the first two terms and factor from the last two terms:

Now we notice that the binomial is common to both terms. We factor the common binomial and get:

**Example 5**

*Factor* .

**Solution**

We factor 3 from the first two terms and factor 4 from the last two terms:

Now factor from both terms: .

Now the polynomial is factored completely.

## Factor Quadratic Trinomials Where a ≠ 1

Factoring by grouping is a very useful method for factoring quadratic trinomials of the form , where .

A quadratic like this doesn’t factor as , so it’s not as simple as looking for two numbers that multiply to and add up to . Instead, we also have to take into account the coefficient in the first term.

To factor a quadratic polynomial where , we follow these steps:

- We find the product .
- We look for two numbers that multiply to and add up to .
- We rewrite the middle term using the two numbers we just found.
- We factor the expression by grouping.

Let’s apply this method to the following examples.

**Example 6**

*Factor the following quadratic trinomials by grouping.*

a)

b)

c)

**Solution**

Let’s follow the steps outlined above:

a)

*Step 1:*

*Step 2:* The number 12 can be written as a product of two numbers in any of these ways:

*Step 3:* Re-write the middle term: , so the problem becomes:

*Step 4:* Factor an from the first two terms and a 2 from the last two terms:

Now factor the common binomial :

To check if this is correct we multiply :

The solution checks out.

b)

*Step 1:*

*Step 2:* The number 24 can be written as a product of two numbers in any of these ways:

*Step 3:* Re-write the middle term: , so the problem becomes:

*Step 4:* Factor by grouping: factor a from the first two terms and a -4 from the last two terms:

Now factor the common binomial :

c)

*Step 1:*

*Step 2:* The number 5 can be written as a product of two numbers in any of these ways:

*Step 3:* Re-write the middle term: , so the problem becomes:

*Step 4:* Factor by grouping: factor an from the first two terms and from the last two terms:

Now factor the common binomial :

## Solve Real-World Problems Using Polynomial Equations

Now that we know most of the factoring strategies for quadratic polynomials, we can apply these methods to solving real world problems.

**Example 7**

*One leg of a right triangle is 3 feet longer than the other leg. The hypotenuse is 15 feet. Find the dimensions of the triangle.*

**Solution**

Let the length of the short leg of the triangle; then the other leg will measure .

Use the Pythagorean Theorem: , where and are the lengths of the legs and is the length of the hypotenuse. When we substitute the values from the diagram, we get .

In order to solve this equation, we need to get the polynomial in standard form. We must first distribute, collect like terms and **rewrite** in the form “polynomial = 0.”

**Factor** out the common monomial:

To factor the trinomial inside the parentheses, we need two numbers that multiply to -108 and add to 3. It would take a long time to go through all the options, so let’s start by trying some of the bigger factors:

We factor the expression as .

**Set each term equal to zero** and **solve:**

It makes no sense to have a negative answer for the length of a side of the triangle, so the answer must be . That means **the short leg is 9 feet and the long leg is 12 feet.**

**Check:** , so the answer checks.

**Example 8**

*The product of two positive numbers is 60. Find the two numbers if one numbers is 4 more than the other.*

**Solution**

Let one of the numbers; then is the other number.

The product of these two numbers is 60, so we can write the equation .

In order to solve we must write the polynomial in standard form. Distribute, collect like terms and **rewrite:**

**Factor** by finding two numbers that multiply to -60 and add to 4. List some numbers that multiply to -60:

The expression factors as .

**Set each term equal to zero** and **solve:**

Since we are looking for positive numbers, the answer must be . **One number is 6, and the other number is 10.**

**Check:** , so the answer checks.

**Example 9**

*A rectangle has sides of length and . What is if the area of the rectangle is 48?*

**Solution**

Make a sketch of this situation:

Using the formula Area = length width, we have .

In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and **rewrite:**

**Factor** by finding two numbers that multiply to -63 and add to 2. List some numbers that multiply to -63:

The expression factors as .

**Set each term equal to zero** and **solve:**

Since we are looking for positive numbers the answer must be . So **the width is** **and the length is** .

**Check:** , so the answer checks.

## Resources

The WTAMU Virtual Math Lab has a detailed page on factoring polynomials here: http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm. This page contains many videos showing example problems being solved.

## Review Questions

Factor completely.

Factor by grouping.

Factor the following quadratic trinomials by grouping.

Solve the following application problems:

- One leg of a right triangle is 7 feet longer than the other leg. The hypotenuse is 13. Find the dimensions of the right triangle.
- A rectangle has sides of and . What value of gives an area of 108?
- The product of two positive numbers is 120. Find the two numbers if one numbers is 7 more than the other.
- A rectangle has a 50-foot diagonal. What are the dimensions of the rectangle if it is 34 feet longer than it is wide?
- Two positive numbers have a sum of 8, and their product is equal to the larger number plus 10. What are the numbers?
- Two positive numbers have a sum of 8, and their product is equal to the smaller number plus 10. What are the numbers?
- Framing Warehouse offers a picture framing service. The cost for framing a picture is made up of two parts: glass costs $1 per square foot and the frame costs $2 per foot. If the frame has to be a square, what size picture can you get framed for $20?