Factor a Trinomial with Leading Coefficient = 1

Learning Outcomes

Factor a trinomial with leading coefficient $= 1$

Trinomials are polynomials with three terms. We are going to show you a method for factoring a trinomial whose leading coefficient is $1$ . Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that trinomials can be factored. The trinomial ${x}^{2}+5x+6$ has a GCF of $1$ , but it can be written as the product of the factors $\left(x+2\right)$ and $\left(x+3\right)$ .

Recall how to use the distributive property to multiply two binomials:

$\left(x+2\right)\left(x+3\right) = x^2+3x+2x+6=x^2+5x+6$

We can reverse the distributive property and return $x^2+5x+6\text{ to }\left(x+2\right)\left(x+3\right)$ by finding two numbers with a product of $6$ and a sum of $5$ .

Factoring a Trinomial with Leading Coefficient 1

In general, for a trinomial of the form ${x}^{2}+bx+c$ , you can factor a trinomial with leading coefficient $1$ by finding two numbers, $p$ and $q$ whose product is c and whose sum is b.

Let us put this idea to practice with the following example.

Example

Factor ${x}^{2}+2x - 15$ .

Show Solution

We have a trinomial with leading coefficient $1,b=2$ , and $c=-15$ . We need to find two numbers with a product of $-15$ and a sum of $2$ . In the table, we list factors until we find a pair with the desired sum.

Factors of $-15$	Sum of Factors
$1,-15$	$-14$
$-1,15$	$14$
$3,-5$	$-2$
$-3,5$	$2$

Now that we have identified $p$ and $q$ as $-3$ and $5$ , write the factored form as $\left(x - 3\right)\left(x+5\right)$ .

In the following video, we present two more examples of factoring a trinomial with a leading coefficient of 1.

To summarize our process, consider the following steps:

How To: Given a trinomial in the form ${x}^{2}+bx+c$ , factor it

List factors of $c$ .
Find $p$ and $q$ , a pair of factors of $c$ with a sum of $b$ .
Write the factored expression $\left(x+p\right)\left(x+q\right)$ .

We will now show an example where the trinomial has a negative c term. Pay attention to the signs of the numbers that are considered for p and q.

In our next example, we show that when c is negative, either p or q will be negative.

Example

Factor $x^{2}+x–12$ .

Show Solution

Consider all the combinations of numbers whose product is $-12$ and list their sum.

Factors whose product is $−12$	Sum of the factors
$1\cdot−12=−12$	$1+−12=−11$
$2\cdot−6=−12$	$2+−6=−4$
$3\cdot−4=−12$	$3+−4=−1$
$4\cdot−3=−12$	$4+−3=1$
$6\cdot−2=−12$	$6+−2=4$
$12\cdot−1=−12$	$12+−1=11$

Choose the values whose sum is $+1$ : $p=4$ and $q=−3$ , and place them into a product of binomials.

$\left(x+4\right)\left(x-3\right)$

Think About It

Which property of multiplication can be used to describe why $\left(x+4\right)\left(x-3\right) =\left(x-3\right)\left(x+4\right)$ . Use the textbox below to write down your ideas before you look at the answer.

Show Solution

The commutative property of multiplication states that factors may be multiplied in any order without affecting the product.

a \cdot b = b \cdot a

$a\cdot b=b\cdot a$

In our last example, we will show how to factor a trinomial whose b term is negative.

Example

Factor ${x}^{2}-7x+6$ .

Show Solution

List the factors of $6$ . Note that the b term is negative, so we will need to consider negative numbers in our list.

Factors of $6$	Sum of Factors
$1,6$	$7$
$2, 3$	$5$
$-1, -6$	$-7$
$-2, -3$	$-5$

Choose the pair that sum to $-7$ , which is $-1, -6$

Write the pair as constant terms in a product of binomials.

$\left(x-1\right)\left(x-6\right)$

In the last example, the b term was negative and the c term was positive. This will always mean that if it can be factored, p and q will both be negative.

Think About It

Can every trinomial be factored as a product of binomials?

Mathematicians often use a counterexample to prove or disprove a question. A counterexample means you provide an example where a proposed rule or definition is not true. Can you create a trinomial with leading coefficient $1$ that cannot be factored as a product of binomials?

Use the textbox below to write your ideas.

Show Solution

Can every trinomial be factored as a product of binomials? No. Some polynomials cannot be factored. These polynomials are said to be prime.

A counterexample would be: $x^2+3x+7$

Module 9: Factoring