4.3 Factoring a Trinomial with Leading Coefficient 1
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].
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].
Question & Answer
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 1: Factoring a Trinomial with Leading Coefficient 1
- Factor [latex]x^2+2x−15[/latex].
- Factor [latex]x^2−7x+6[/latex].
Analysis
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].
Question & Answer
Does the order of the factors matter?
No. Multiplication is commutative, so the order of the factors does not matter.
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