Factor a Trinomial with Leading Coefficient = 1
Learning Outcomes
- Factor a trinomial with leading coefficient [latex]= 1[/latex]
[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.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] |
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].
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] |
[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.
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] |
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.