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Study Guides > Intermediate Algebra

Read: Factor by Grouping

Learning Objectives

  • Factor a trinomial with leading coefficient other than 11 using grouping
Trinomials with leading coefficients other than 11 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial 2x2+5x+32{x}^{2}+5x+3 can be rewritten as (2x+3)(x+1)\left(2x+3\right)\left(x+1\right) using this process. We begin by rewriting the original expression as 2x2+2x+3x+32{x}^{2}+2x+3x+3 and then factor each portion of the expression to obtain 2x(x+1)+3(x+1)2x\left(x+1\right)+3\left(x+1\right). We then pull out the GCF of (x+1)\left(x+1\right) to find the factored expression. The first step in this process is to figure out what two numbers to use to re-write the x term as the sum of two new terms. Making a table to keep track of your work is helpful. We are looking for two numbers with a product of 66 and a sum of 55
Factors of 23=62\cdot3=6 Sum of Factors
1,61,6 77
1,6-1,-6 7-7
2,32,3 55
2,3-2,-3 5-5
  The pair p=2, and q=3p=2,\text{ and }q=3 will give the correct x term, so we will rewrite it using the new factors:

2x2+5x+3=2x2+2x+3x+32{x}^{2}+5x+3=2x^2+2x+3x+3

Now we can group the polynomial into two binomials.

2x2+2x+3x+3=(2x2+2x)+(3x+3)2x^2+2x+3x+3=(2x^2+2x)+(3x+3)

Identify the GCF of each binomial.

2x2x is the GCF of (2x2+2x)(2x^2+2x) and 33 is the GCF of (3x+3)(3x+3), use this to rewrite the polynomial:

(2x2+2x)+(3x+3)=2x(x+1)+3(x+1)(2x^2+2x)+(3x+3)=2x(x+1)+3(x+1)

Note how we leave the signs in the binomials and the addition that joins them, be careful with signs when you factor out the GCF. The GCF of our new polynomial is (x+1)(x+1), we factor this out as well:

2x(x+1)+3(x+1)=(x+1)(2x+3)2x(x+1)+3(x+1)=(x+1)(2x+3).

Sometimes it helps visually to write the polynomial this way (x+1)2x+(x+1)3(x+1)2x+(x+1)3 before you factor out the GCF. This is purely a matter of preference, multiplication is commutative, so order doesn't matter.

 A General Note: Factor by Grouping

To factor a trinomial in the form ax2+bx+ca{x}^{2}+bx+c by grouping, we find two numbers with a product of acac and a sum of bb. We use these numbers to divide the xx term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.

Example

Factor 5x2+7x65{x}^{2}+7x - 6 by grouping.

Answer: We have a trinomial with a=5,b=7a=5,b=7, and c=6c=-6. First, determine ac=30ac=-30. We need to find two numbers with a product of 30-30 and a sum of 77. In the table, we list factors until we find a pair with the desired sum.

Factors of 30-30 Sum of Factors
1,301,-30 29-29
1,30-1,30 2929
2,152,-15 13-13
2,15-2,15 1313
3,103,-10 7-7
3,10-3,10 77
So p=3p=-3 and q=10q=10.
5x23x+10x6Rewrite the original expression as ax2+px+qx+c.x(5x3)+2(5x3)Factor out the GCF of each part.(5x3)(x+2)Factor out the GCF  of the expression.\begin{array}{cc}5{x}^{2}-3x+10x - 6 \hfill & \text{Rewrite the original expression as }a{x}^{2}+px+qx+c.\hfill \\ x\left(5x - 3\right)+2\left(5x - 3\right)\hfill & \text{Factor out the GCF of each part}.\hfill \\ \left(5x - 3\right)\left(x+2\right)\hfill & \text{Factor out the GCF}\text{ }\text{ of the expression}.\hfill \end{array}

Analysis of the Solution

We can check our work by multiplying. Use FOIL to confirm that (5x3)(x+2)=5x2+7x6\left(5x - 3\right)\left(x+2\right)=5{x}^{2}+7x - 6.
We can summarize our process in the following way:

Given a trinomial in the form ax2+bx+ca{x}^{2}+bx+c, factor by grouping.

  1. List factors of acac.
  2. Find pp and qq, a pair of factors of acac with a sum of bb.
  3. Rewrite the original expression as ax2+px+qx+ca{x}^{2}+px+qx+c.
  4. Pull out the GCF of ax2+pxa{x}^{2}+px.
  5. Pull out the GCF of qx+cqx+c.
  6. Factor out the GCF of the expression.
In the following video we present one more example of factoring a trinomial whose leading coefficient is not 1 using the grouping method. https://youtu.be/agDaQ_cZnNc Factoring trinomials whose leading coefficient is not 11 becomes quick and kind of fun once you get the idea.  Give the next example a try on your own before you look at the solution.

We will show two more examples so you can become acquainted with the variety of possible outcomes for factoring this type of trinomial.

Example

Factor 2x2+9x+92{x}^{2}+9x+9.

Answer: Find two numbers p, q such that pq=18p\cdot{q}=18, and p+q=9p + q = 999 and 1818 are both positive, so we will only consider positive factors.

Factors of 29=182\cdot9=18 Sum of Factors
1,181, 18 1919
3,63,6 99
We can stop because we have found our factors. Rewrite the original expression, and group.

2x2+3x+6x+9=(2x2+3x)+(6x+9)2x^2+3x+6x+9=(2x^2+3x)+(6x+9)

Factor out the GCF of each binomial, and write as a product of two binomials:

(2x2+3x)+(6x+9)=x(2x+3)+3(2x+3)=(x+3)(2x+3)(2x^2+3x)+(6x+9)=x(2x+3)+3(2x+3)=(x+3)(2x+3)

2x2+9x+9=(x+3)(2x+3)2{x}^{2}+9x+9=(x+3)(2x+3)

Here is an example where the x term is positive and c is negative.

Example

Factor 6x2+x16{x}^{2}+x - 1.

Answer:  

Factors of 61=66\cdot-1=-6 Sum of Factors
1,6-1,6 55
1,61,-6 5-5
2,3-2,3 11
We can stop because we have found our factors. Rewrite the original expression, and group.

6x2+x1=6x22x+3x16{x}^{2}+x - 1=6x^2-2x+3x-1

Factor out the GCF of each binomial, and write as a product of two binomials:

(6x22x)+(3x1)=2x(3x1)+1(3x1)=(2x+1)(3x1)(6x^2-2x)+(3x-1)=2x(3x-1)+1(3x-1)=(2x+1)(3x-1)

6x2+x1=(2x+1)(3x1)6{x}^{2}+x - 1=(2x+1)(3x-1)

In the following video example, we will factor a trinomial whose leading term is negative. https://youtu.be/zDAMjdBfkDs For our last example, you will see that sometimes, you will encounter polynomials that, despite your best efforts, cannot be factored into the product of two binomials.

Example

Factor 7x216x57x^{2}-16x–5.

Answer: Find p,qp, q such that pq=35 and p+q=16p\cdot{q}=-35\text{ and }p+q=-16

Factors of 75=357\cdot{-5}=-35 Sum of Factors
1,35-1, 35 3434
1,351, -35 34-34
5,7-5, 7 22
7,5-7,5 2-2
 

Answer

Cannot be factored. None of the factors add up to 16-16

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