Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial [latex]{x}^{2}+5x+6[/latex] has a GCF of 1, but it can be written as the product of the factors [latex]\left(x+2\right)[/latex] and [latex]\left(x+3\right)[/latex].
Trinomials of the form [latex]{x}^{2}+bx+c[/latex] can be factored by finding two numbers with a product of [latex]c[/latex] and a sum of [latex]b[/latex]. The trinomial [latex]{x}^{2}+10x+16[/latex], for example, can be factored using the numbers [latex]2[/latex] and [latex]8[/latex] because the product of those numbers is [latex]16[/latex] and their sum is [latex]10[/latex]. The trinomial can be rewritten as the product of [latex]\left(x+2\right)[/latex] and [latex]\left(x+8\right)[/latex].
A General Note: Factoring a Trinomial with Leading Coefficient 1
A trinomial of the form [latex]{x}^{2}+bx+c[/latex] can be written in factored form as [latex]\left(x+p\right)\left(x+q\right)[/latex] where [latex]pq=c[/latex] and [latex]p+q=b[/latex].
Q & A
Can every trinomial be factored as a product of binomials?
No. Some polynomials cannot be factored. These polynomials are said to be prime.
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 2: Factoring a Trinomial with Leading Coefficient 1
Factor [latex]{x}^{2}+2x - 15[/latex].
Solution
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] | 14 |
| [latex]3,-5[/latex] | [latex]-2[/latex] |
| [latex]-3,5[/latex] | 2 |
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].
Analysis of the Solution
We can check our work by multiplying. Use FOIL to confirm that [latex]\left(x - 3\right)\left(x+5\right)={x}^{2}+2x - 15[/latex].
Q & A
Does the order of the factors matter?
No. Multiplication is commutative, so the order of the factors does not matter.