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Factoring Polynomials by Grouping

Factoring by grouping is often used to factor a four-term polynomial such as 10x2 + 35x - 6xy - 21y.

 

Procedure — To Factor a Polynomial by Grouping

Step 1 Factor each term.

Step 2 Group terms with common factors.

Step 3 In each group, factor out the GCF.

Step 4 Factor out the GCF of the polynomial from Step 3.

 

Example 1

Factor: 10x2 + 35x - 6xy - 21y

Solution

Step 1 Factor each term.

 

The GCF of the first two terms is 5x.

 

 

The GCF of the second two terms is 3y.

10x2 = 2 · 5 · x · x

35x = 5 · 7 · x

 

-6xy = -1 · 2 · 3 · x · y

-21y = -1 · 3 · 7 · y

 

Step 2 Group terms with common factors. (10x2 + 35x) + (-6xy - 21y)

= (5x · 2x + 5x · 7) + (-1 · 3y · 2x + (-1 · 3y) · 7)

Step 3 In each group, factor out the GCF. = 5x(2x + 7) + (-3y)(2x + 7)
Step 4 Factor out the GCF of the polynomial from Step 3.

The binomial (2x + 7) is common to both groups.

 

= (2x + 7)(5x - 3y)

So, the factorization is (2x + 7)(5x - 3y).

Note:

Often there is more than one way to form two groups of two factors so that each has at least one common factor.

For 10x2 + 35x - 6xy - 21y, we could also group like this:

(10x2 - 6xy) + (35x - 21y)

2x(5x - 3y) + 7(5x - 3y)

(5x - 3y)(2x + 7)

 

Example 2

Factor: 6x2 + 3xy + 2x + y.

Solution

Step 1 Factor each term.

 

The GCF of the first two terms is 3x.

 

 

The GCF of the second two terms is 1.

6x2 = 2 · 3 · x · x

3xy = 3 · x · y

 

2x = 2 · x = 1 · 2 · x

y = 1 · y

 

Step 2 Group terms with common factors. (6x2 + 3xy) + (2x + y)

= (3x · 2x + 3x · y) + (1 · 2x + 1 · y)

Step 3 In each group, factor out the GCF. = 3x(2x + y) + 1(2x + y)
Step 4 Factor out the GCF of the polynomial from Step 3.

The binomial (2x + 7) is common to both groups.

 

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

Thus, the factorization is (2x + y)(3x + 1).

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