Factor a Trinomial with Leading Coefficient = 1

Learning Outcomes

Factor a trinomial with leading coefficient [latex]= 1[/latex]

Trinomials are polynomials with three terms. We are going to show you a method for factoring a trinomial whose leading coefficient is [latex]1[/latex]. 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 [latex]{x}^{2}+5x+6[/latex] has a GCF of [latex]1[/latex], but it can be written as the product of the factors [latex]\left(x+2\right)[/latex] and [latex]\left(x+3\right)[/latex].

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

[latex]\left(x+2\right)\left(x+3\right) = x^2+3x+2x+6=x^2+5x+6[/latex]

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

Factoring a Trinomial with Leading Coefficient 1

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

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

Example

Factor [latex]{x}^{2}+2x - 15[/latex].

Show Solution

Factors of [latex]-15[/latex]	Sum of Factors
[latex]1,-15[/latex]	[latex]-14[/latex]
[latex]-1,15[/latex]	[latex]14[/latex]
[latex]3,-5[/latex]	[latex]-2[/latex]
[latex]-3,5[/latex]	[latex]2[/latex]

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 [latex]{x}^{2}+bx+c[/latex], factor it

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

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 [latex]x^{2}+x–12[/latex].

Show Solution

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

Think About It

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

Show Solution

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

Example

Factor [latex]{x}^{2}-7x+6[/latex].

Show Solution

Factors of [latex]6[/latex]	Sum of Factors
[latex]1,6[/latex]	[latex]7[/latex]
[latex]2, 3[/latex]	[latex]5[/latex]
[latex]-1, -6[/latex]	[latex]-7[/latex]
[latex]-2, -3[/latex]	[latex]-5[/latex]

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 [latex]1[/latex] that cannot be factored as a product of binomials?

Use the textbox below to write your ideas.

Show Solution

Module 9: Factoring