Polynomial Equations: Background You’ll Need 2

  • Factor trinomials where a is not 1

Factoring by Grouping (Factoring a Trinomial with Leading Coefficient of Not 1)

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the [latex]x[/latex] term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression.

Factoring Trinomial (Leading Coefficient [latex]\ne 1[/latex])

Factoring by grouping is a method used to decompose a trinomial of the form [latex]ax^2+bx+c[/latex] into a product of two binomials.

 

This method involves finding two numbers that combine to give the product of the leading coefficient and the constant term ([latex]a \times c[/latex]) and the sum of the middle coefficient ([latex]b[/latex]). We use these numbers to divide the [latex]x[/latex] term into the sum of two terms and factor each portion of the expression separately then factor out the GCF of the entire expression.

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

  1. List factors of [latex]a \times c[/latex].
  2. Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]a \times c[/latex] with a sum of [latex]b[/latex].
  3. Rewrite the original expression as [latex]a{x}^{2}+px+qx+c[/latex].
  4. Pull out the GCF of [latex]a{x}^{2}+px[/latex].
  5. Pull out the GCF of [latex]qx+c[/latex].
  6. Factor out the GCF of the expression.
Factor [latex]5{x}^{2}+7x - 6[/latex].

Solution

  • List factors of [latex]a \times c[/latex].
    • Calculate [latex]a \times c[/latex]: [latex]a = 5[/latex] and [latex]c = -6[/latex], so [latex]a \times c = -30[/latex].
    • Factors of [latex]-30[/latex]:
      Factors of [latex]-30[/latex] Sum of Factors
      [latex]1,-30[/latex] [latex]-29[/latex]
      [latex]-1,30[/latex] [latex]29[/latex]
      [latex]2,-15[/latex] [latex]-13[/latex]
      [latex]-2,15[/latex] [latex]13[/latex]
      [latex]3,-10[/latex] [latex]-7[/latex]
      [latex]-3,10[/latex] [latex]7[/latex]
  • Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]a \times c[/latex] with a sum of [latex]b[/latex].
    • Based on the table above, the correct pair is [latex]p = -3[/latex] and [latex]q = 10[/latex].
  • Rewrite the original expression as [latex]a{x}^{2}+px+qx+c[/latex].

[latex]5{x}^{2}+7x - 6 = 5{x}^{2} -3x+10x-6[/latex]

  • Group the first 2 terms and the last 2 terms. Then, pull out the GCF of each group.

[latex]5{x}^{2}+7x - 6 = (5{x}^{2} -3x)+(10x-6) = x(5x-3)+2(5x-3)[/latex]

  • Factor out the GCF of the expression.

[latex]5{x}^{2}+7x - 6 = (x+2)(5x-3)[/latex]

Factor the following.

  1. [latex]2{x}^{2}+9x+9[/latex]
  2. [latex]6{x}^{2}+x - 1[/latex]