# 9.5: Factoring Quadratic Expressions

## Learning Objectives

- Write quadratic equations in standard form.
- Factor quadratic expressions for different coefficient values.
- Factor when .

## Write Quadratic Expressions in Standard Form

**Quadratic polynomials** are polynomials of degree. The standard form of a quadratic polynomial is written as

Here and stand for constant numbers. Factoring these polynomials depends on the values of these constants. In this section, we will learn how to factor quadratic polynomials for different values of and . In the last section, we factored common monomials, so you already know how to factor quadratic polynomials where .

For example for the quadratic , the common factor is and this expression is factored as . When all the coefficients are not zero these expressions are also called **Quadratic Trinomials**, since they are polynomials with three terms.

## Factor when a = 1, b is Positive, and c is Positive

Let’s first consider the case where is positive and is positive. The quadratic trinomials will take the following form.

You know from multiplying binomials that when you multiply two factors you obtain a quadratic polynomial. Let’s multiply this and see what happens. We use The Distributive Property.

To simplify this polynomial we would combine the like terms in the middle by adding them.

To factor we need to do this process in reverse.

This means that we need to find two numbers and where

To factor , the answer is the product of two parentheses.

so that and

Let’s try some specific examples.

**Example 1**

*Factor*

**Solution** We are looking for an answer that is a product of two binomials in parentheses.

To fill in the blanks, we want two numbers and that multiply to 6 and add to 5. A good strategy is to list the possible ways we can multiply two numbers to give us 6 and then see which of these pairs of numbers add to 5. The number six can be written as the product of.

So the answer is .

We can check to see if this is correct by multiplying .

**The answer checks out.**

**Example 2**

*Factor*

**Solution**

We are looking for an answer that is a product of two parentheses .

The number 12 can be written as the product of the following numbers.

The answer is .

**Example 3**

*Factor* .

**Solution**

We are looking for an answer that is a product of the two parentheses .

The number 12 can be written as the product of the following numbers.

The answer is

**Example 4**

*Factor* .

**Solution**

We are looking for an answer that is a product of the two parentheses .

The number 36 can be written as the product of the following numbers.

The answer is .

## Factor when a = 1, b is Negative and c is Positive

Now let’s see how this method works if the middle coefficient is negative.

**Example 5**

*Factor*

**Solution**

We are looking for an answer that is a product of the two parentheses .

The number 8 can be written as the product of the following numbers.

and Notice that these are two different choices.

** But also**,

** But also**,

The answer is

We can check to see if this is correct by multiplying .

The answer checks out.

**Example 6**

*Factor*

**Solution**

We are looking for an answer that is a product of two parentheses: .

The number 16 can be written as the product of the following numbers:

The answer is .

## Factor when a = 1 and c is Negative

Now let’s see how this method works if the constant term is negative.

**Example 7**

*Factor*

**Solution**

We are looking for an answer that is a product of two parentheses .

In this case, we must take the negative sign into account. The number -15 can be written as the product of the following numbers.

**And also**,

The answer is .

We can check to see if this is correct by multiplying .

The answer checks out.

**Example 8**

*Factor*

**Solution**

We are looking for an answer that is a product of two parentheses .

The number -24 can be written as the product of the following numbers.

The answer is

**Example 9**

Factor

**Solution**

We are looking for an answer that is a product of two parentheses

The number -35 can be written as the product of the following numbers:

The answer is .

## Factor when a = - 1

When , the best strategy is to factor the common factor of -1 from all the terms in the quadratic polynomial. Then, you can apply the methods you have learned so far in this section to find the missing factors.

**Example 10**

*Factor*

**Solution**

First factor the common factor of -1 from each term in the trinomial. Factoring -1 changes the signs of each term in the expression.

We are looking for an answer that is a product of two parentheses

Now our job is to factor .

The number -6 can be written as the product of the following numbers.

The answer is .

**To Summarize,**

A quadratic of the form factors as a product of two parenthesis .

- If and are positive then both and are positive
- Example factors as .

- If is negative and is positive then both and are negative.
- Example factors as .

- If is negative then either is positive and is negative or vice-versa
- Example factors as .
- Example factors as .

- If , factor a common factor of -1 from each term in the trinomial and then factor as usual. The answer will have the form .
- Example factors as .

## Review Questions

Factor the following quadratic polynomials.