Factoring by Grouping
Learning Outcome
- Apply an algorithm to rewrite a trinomial as a four term polynomial and factor
- Use factoring by grouping to factor a trinomial
- Factor trinomials of the form [latex]a{x}^{2}+bx+c[/latex]
Factoring Trinomials in the form [latex]ax^{2}+bx+c[/latex]
To factor a trinomial in the form [latex]ax^{2}+bx+c[/latex], find two integers, r and s, whose sum is b and whose product is ac.[latex]\begin{array}{l}r\cdot{s}=a\cdot{c}\\r+s=b\end{array}[/latex]
Rewrite the trinomial as [latex]ax^{2}+rx+sx+c[/latex] and then use grouping and the distributive property to factor the polynomial.| Factors of [latex]2\cdot3=6[/latex] | Sum of Factors |
|---|---|
| [latex]1,6[/latex] | [latex]7[/latex] |
| [latex]-1,-6[/latex] | [latex]-7[/latex] |
| [latex]2,3[/latex] | [latex]5[/latex] |
| [latex]-2,-3[/latex] | [latex]-5[/latex] |
[latex]2{x}^{2}+5x+3=2x^2+2x+3x+3[/latex]
Now we can group the polynomial into two binomials.
[latex]2x^2+2x+3x+3=(2x^2+2x)+(3x+3)[/latex]
Identify the GCF of each binomial.
[latex]2x[/latex] is the GCF of [latex](2x^2+2x)[/latex] and [latex]3[/latex] is the GCF of [latex](3x+3)[/latex]. Use this to rewrite the polynomial:
[latex](2x^2+2x)+(3x+3)=2x(x+1)+3(x+1)[/latex]
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 [latex](x+1)[/latex]. We factor this out as well:
[latex]2x(x+1)+3(x+1)=(x+1)(2x+3)[/latex].
Sometimes it helps visually to write the polynomial as [latex](x+1)2x+(x+1)3[/latex] before you factor out the GCF. This is purely a matter of preference. Multiplication is commutative, so order does not matter.
Now let’s see how this strategy works for factoring [latex]6z^{2}+11z+4[/latex]. In this trinomial, [latex]a=6[/latex], [latex]b=11[/latex], and [latex]c=4[/latex]. According to the strategy, you need to find two factors, r and s, whose sum is [latex]b=11[/latex] and whose product is [latex]a\cdot{c}=6\cdot4=24[/latex]. You can make a chart to organize the possible factor combinations. (Notice that this chart only has positive numbers. Since ac is positive and b is positive, you can be certain that the two factors you're looking for are also positive numbers.)| Factors whose product is [latex]24[/latex] | Sum of the factors |
|---|---|
| [latex]1\cdot24=24[/latex] | [latex]1+24=25[/latex] |
| [latex]2\cdot12=24[/latex] | [latex]2+12=14[/latex] |
| [latex]3\cdot8=24[/latex] | [latex]3+8=11[/latex] |
| [latex]4\cdot6=24[/latex] | [latex]4+6=10[/latex] |
Example
Factor [latex]6z^{2}+11z+4[/latex].Answer: Rewrite the middle term, [latex]11z[/latex], as [latex]3z + 8z[/latex] (from the chart above.)
[latex]6z^{2}+3z+8z+4[/latex]
Group pairs. Use grouping to consider the terms in pairs.[latex]\left(6z^{2}+3z\right)+\left(8z+4\right)[/latex]
Factor [latex]3[/latex]z out of the first group and [latex]4[/latex] out of the second group.[latex]3z\left(2z+1\right)+4\left(2z+1\right)[/latex]
Factor out [latex]\left(2z+1\right)[/latex].[latex]\left(2z+1\right)\left(3z+4\right)[/latex]
Answer
[latex-display]\left(2z+1\right)\left(3z+4\right)[/latex-display]Example
Factor [latex]5{x}^{2}+7x - 6[/latex] by grouping.Answer: 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.
| 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] |
We can summarize our process in the following way:
In the following video, we present another example of factoring a trinomial using grouping. In this example, the middle term, b, is negative. Note how having a negative middle term and a positive c term influence the options for r and s when factoring.
https://youtu.be/agDaQ_cZnNc
Using this strategy, factoring trinomials becomes quick and kind of fun once you get the idea. Give the next examples a try on your own before you look at the solution.
How To: Given a trinomial in the form [latex]a{x}^{2}+bx+c[/latex], factor by grouping
- List factors of [latex]ac[/latex].
- Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]ac[/latex] with a sum of [latex]b[/latex].
- Rewrite the original expression as [latex]a{x}^{2}+px+qx+c[/latex].
- Pull out the GCF of [latex]a{x}^{2}+px[/latex].
- Pull out the GCF of [latex]qx+c[/latex].
- Factor out the GCF of the expression.
Try It
[ohm_question]115262[/ohm_question]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 [latex]2{x}^{2}+9x+9[/latex].Answer: Find two numbers p, q such that [latex]p\cdot{q}=18[/latex] and [latex]p + q = 9[/latex]. [latex]9[/latex] and [latex]18[/latex] are both positive, so we will only consider positive factors.
| Factors of [latex]2\cdot9=18[/latex] | Sum of Factors |
|---|---|
| [latex]1, 18[/latex] | [latex]19[/latex] |
| [latex]3,6[/latex] | [latex]9[/latex] |
[latex]2x^2+3x+6x+9=(2x^2+3x)+(6x+9)[/latex]
Factor out the GCF of each binomial and write as a product of two binomials:[latex](2x^2+3x)+(6x+9)=x(2x+3)+3(2x+3)=(x+3)(2x+3)[/latex]
[latex-display]2{x}^{2}+9x+9=(x+3)(2x+3)[/latex-display]Example
Factor [latex]6{x}^{2}+x - 1[/latex].Answer:
| Factors of [latex]6\cdot-1=-6[/latex] | Sum of Factors |
|---|---|
| [latex]-1,6[/latex] | [latex]5[/latex] |
| [latex]1,-6[/latex] | [latex]-5[/latex] |
| [latex]-2,3[/latex] | [latex]1[/latex] |
[latex]6{x}^{2}+x - 1=6x^2-2x+3x-1[/latex]
Factor out the GCF of each binomial and write as a product of two binomials:[latex](6x^2-2x)+(3x-1)=2x(3x-1)+1(3x-1)=(2x+1)(3x-1)[/latex]
[latex-display]6{x}^{2}+x - 1=(2x+1)(3x-1)[/latex-display]Prime Trinomials
Before going any further, it is worth mentioning that not all trinomials can be factored using integer pairs. Take the trinomial [latex]2z^{2}+35z+7[/latex], for instance. Can you think of two integers whose sum is [latex]b=35[/latex] and whose product is [latex]a\cdot{c}=2\cdot7=14[/latex]? There are none! This type of trinomial, which cannot be factored using integers, is called a prime trinomial. 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 [latex]7x^{2}-16x–5[/latex].Answer: Find [latex]p, q[/latex] such that [latex]p\cdot{q}=-35\text{ and }p+q=-16[/latex]
| Factors of [latex]7\cdot{-5}=-35[/latex] | Sum of Factors |
|---|---|
| [latex]-1, 35[/latex] | [latex]34[/latex] |
| [latex]1, -35[/latex] | [latex]-34[/latex] |
| [latex]-5, 7[/latex] | [latex]2[/latex] |
| [latex]-7,5[/latex] | [latex]-2[/latex] |
Factoring out a GCF
Now that you are familiar with the grouping method for factoring, we will now start to look at some strategies that are useful for making factoring easier. Your first step when factoring should always be to look for common factors for the three terms. Consider the examples below.| Trinomial | Factor out Common Factor | Factored |
|---|---|---|
| [latex]2x^{2}+10x+12[/latex] | [latex]2(x^{2}+5x+6)[/latex] | [latex]2\left(x+2\right)\left(x+3\right)[/latex] |
| [latex]−5a^{2}−15a−10[/latex] | [latex]−5(a^{2}+3a+2)[/latex] | [latex]−5\left(a+2\right)\left(a+1\right)[/latex] |
| [latex]c^{3}–8c^{2}+15c[/latex] | [latex]c\left(c^{2}–8c+15\right)[/latex] | [latex]c\left(c–5\right)\left(c–3\right)[/latex] |
| [latex]y^{4}–9y^{3}–10y^{2}[/latex] | [latex]y^{2}\left(y^{2}–9y–10\right)[/latex] | [latex]y^{2}\left(y–10\right)\left(y+1\right)[/latex] |
Example
Factor [latex]3x^{3}–3x^{2}–90x[/latex].Answer: Since 3 is a common factor for the three terms, factor out the 3.
[latex]3\left(x^{3}–x^{2}–30x\right)[/latex]
x is also a common factor, so factor out x.[latex]3x\left(x^{2}–x–30\right)[/latex]
Now you can factor the trinomial [latex]x^{2}–x–30[/latex]. To find r and s, identify two numbers whose product is [latex]−30[/latex] and whose sum is [latex]−1[/latex]. The pair of factors is [latex]−6[/latex] and [latex]5[/latex]. So replace [latex]–x[/latex] with [latex]−6x+5x[/latex].[latex]3x\left(x^{2}–6x+5x–30\right)[/latex]
Use grouping to consider the terms in pairs.[latex]3x\left[\left(x^{2}–6x\right)+\left(5x–30\right)\right][/latex]
Factor x out of the first group and factor [latex]5[/latex] out of the second group.[latex]3x\left[\left(x\left(x–6\right)\right)+5\left(x–6\right)\right][/latex]
Then factor out [latex]x–6[/latex].[latex]3x\left(x–6\right)\left(x+5\right)[/latex]
Answer
[latex-display]3x\left(x–6\right)\left(x+5\right)[/latex-display]Factoring out a Negative
In some situations, a is negative, as in [latex]−4h^{2}+11h+3[/latex]. It often makes sense to factor out [latex]−1[/latex] as the first step in factoring, as doing so will change the sign of [latex]ax^{2}[/latex] from negative to positive, making the remaining trinomial easier to factor.Example
Factor [latex]−4h^{2}+11h+3[/latex]Answer: Factor [latex]−1[/latex] out of the trinomial. Notice that the signs of all three terms have changed.
[latex]−1\left(4h^{2}–11h–3\right)[/latex]
To factor the trinomial, you need to figure out how to rewrite [latex]−11h[/latex]. The product of [latex]rs=4\cdot−3=−12[/latex], and the sum of [latex]rs=−11[/latex].| [latex]r\cdot{s}=−12[/latex] | [latex]r+s=−11[/latex] |
| [latex]−12\cdot1=−12[/latex] | [latex]−12+1=−11[/latex] |
| [latex]−6\cdot2=−12[/latex] | [latex]−6+2=−4[/latex] |
| [latex]−4\cdot3=−12[/latex] | [latex]−4+3=−1[/latex] |
[latex]−1\left(4h^{2}–12h+1h–3\right)[/latex]
Group terms.[latex]−1\left[\left(4h^{2}–12h\right)+\left(1h–3\right)\right][/latex]
Factor out [latex]4[/latex]h from the first pair. The second group cannot be factored further, but you can write it as [latex]+1\left(h–3\right)[/latex] since [latex]+1\left(h–3\right)=\left(h–3\right)[/latex]. This helps with factoring in the next step.[latex]−1\left[4h\left(h–3\right)+1\left(h–3\right)\right][/latex]
Factor out a common factor of [latex]\left(h–3\right)[/latex]. Notice you are left with [latex]\left(h–3\right)\left(4h+1\right)[/latex]; the [latex]+1[/latex] comes from the term [latex]+1\left(h–3\right)[/latex] in the previous step.[latex]−1\left[\left(h–3\right)\left(4h+1\right)\right][/latex]
Answer
[latex-display]−1\left(h–3\right)\left(4h+1\right)[/latex-display]Contribute!
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Licenses & Attributions
CC licensed content, Original
- Factor a Trinomial in the Form ax^2+bx+c Using the Grouping Technique. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Factor a Trinomial in the Form -ax^2+bx+c Using the Grouping Technique. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 12: Factoring, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
