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Study Guides > Intermediate Algebra

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].

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

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

In the following video, we present two more examples of factoring a trinomial with a leading coefficient of 1. https://youtu.be/-SVBVVYVNTM To summarize our process, consider the following steps:

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

  1. List factors of [latex]c[/latex].
  2. Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]c[/latex] with a sum of [latex]b[/latex].
  3. 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].

Answer: Consider all the combinations of numbers whose product is [latex]-12[/latex] and list their sum.

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

[latex]\left(x+4\right)\left(x-3\right)[/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. [practice-area rows="2"][/practice-area]

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

[latex]a\cdot b=b\cdot a[/latex]

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].

Answer: List the factors of [latex]6[/latex]. Note that the b term is negative, so we will need to consider negative numbers in our list.

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]
Choose the pair that sum to [latex]-7[/latex], which is [latex]-1, -6[/latex] Write the pair as constant terms in a product of binomials. [latex-display]\left(x-1\right)\left(x-6\right)[/latex-display]

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. [practice-area rows="2"][/practice-area]

Answer: 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: [latex]x^2+3x+7[/latex]

Licenses & Attributions

CC licensed content, Original

  • Factor a Trinomial Using the Shortcut Method - Form x^2+bx+c. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.

CC licensed content, Shared previously

  • Unit 12: Factoring, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.