Using the Greatest Common Factor and Factor by Grouping
Factor by Grouping
When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.
(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime).
Example 7.15
How to Factor by Grouping
Factor: \(xy+3y+2x+6\).
Solution
| Step 1. Group terms with common factors. | Is there a greatest common factor of all four terms? No, so let's separate the first two terms from the second two. |
\(xy+3y+2x+6\) \( \underbrace{\operatorname{xy+3y}}_{ } \; \; \underbrace{\operatorname{ + 2x+6}}_{ } \) |
| Step 2. Factor out the common factor in each group. | Factor the GCF from the first two terms. Factor the GCF from the second two terms. |
\( y(x+3) \; \; \underbrace{\operatorname{ + 2x+6}}_{ } \) \(y((x+3) + 2(x+3)\) |
| Step 3. Factor the common factor from the expression. | Notice that each term has a common factor of \((x+3)\). Factor out the common factor. |
\(y(x+3)+2(x+3)\) \((x+3)(y+2)\) |
| Step 4. Check. | Multiply \((x+3)(y+2)\). Is the product the original expression? | \( \begin{array}{l} (x+3)(y+2) \\ x y+2 x+3 y+6 \\ x y+3 y+2 x+6 \text{✓} \end{array} \) |
Try It 7.29
Factor: \(x y+8 y+3 x+24\).
Try It 7.30
Factor: \(a b+7 b+8 a+56\).
HOW TO
Factor by grouping.
- Step 1. Group terms with common factors.
- Step 2. Factor out the common factor in each group.
- Step 3. Factor the common factor from the expression.
- Step 4. Check by multiplying the factors.
Example 7.16
Factor: \(x^{2}+3 x-2 x-6\).
Solution
| There is no GCF in all four terms. Separate into two parts. |
\(x^2+3x-2x-6\) \( \underbrace{\operatorname{x^2+3x}}_{ } \; \; \underbrace{\operatorname{ -2x-6}}_{ } \) |
| Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms. |
\( \begin{array}{c} x(x+3)-2(x+3) \\ (x+3)(x-2) \end{array} \) |
| Check on your own by multiplying. |
Try It 7.31
Factor: \(x^{2}+2 x-5 x-10\).
Try It 7.32
Factor: \(y^{2}+4 y-7 y-28\).