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To factor a trinomial in the form [latex]a^+bx+c[/latex] by grouping, we find two numbers with a product of [latex]ac[/latex] and a sum of [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^+bx+c[/latex], factor by grouping.

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

Example 3: Factoring a Trinomial by Grouping

Factor [latex]5^+7x - 6[/latex] by grouping.

Solution

We have a trinomial with [latex]a=5,b=7[/latex], and [latex]c=-6[/latex]. First, determine [latex]ac=-30[/latex]. We need to find two numbers with a product of [latex]-30[/latex] and a sum of [latex]7[/latex]. In the table, we list factors until we find a pair with the desired sum.

second is labeled: Sum of Factors. The entries in the first column are: 1, -30; -1, 30; 2, -15; -2, 15; 3, -10; and -3, 10. The entries in the second column are: -29, 29, -13, 13, -7, and 7.">
Factors of [latex]-30[/latex] Sum of Factors
[latex]1,-30[/latex] [latex]-29[/latex]
[latex]-1,30[/latex] 29
[latex]2,-15[/latex] [latex]-13[/latex]
[latex]-2,15[/latex] 13
[latex]3,-10[/latex] [latex]-7[/latex]
[latex]-3,10[/latex] 7
So [latex]p=-3[/latex] and [latex]q=10[/latex].

Analysis of the Solution

We can check our work by multiplying. Use FOIL to confirm that [latex]\left(5x - 3\right)\left(x+2\right)=5^+7x - 6[/latex].

Try It 3

Factor the following.

a.[latex]2^+9x+9[/latex] b. [latex]6^+x - 1[/latex]

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