Factoring Polynomials: Learn It 2

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]

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]