Polynomial Equations: Background You’ll Need 1

  • Factor trinomials with a = 1

Factoring a Trinomial with Leading Coefficient of 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].

Factoring a Trinomial (Leading Coefficient  [latex]= 1[/latex])

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].

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

  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].
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] [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].

Thus, [latex]{x}^{2}+2x - 15 = \left(x - 3\right)\left(x+5\right)[/latex]