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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 â€” 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 Â· 5 Â· x Â· x 35x = 5 Â· 7 Â· x   -6xy = -1 Â· 2 Â· 3 Â· x Â· y-21y = -1 Â· 3 Â· 7 Â· y Step 2 Group terms with common factors. (10x2 + 35x) + (-6xy - 21y) = (5x Â· 2x + 5x Â· 7) + (-1 Â· 3y Â· 2x + (-1 Â· 3y) Â· 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 Â· 3 Â· x Â· x 3xy = 3 Â· x Â· y   2x = 2 Â· x = 1 Â· 2 Â· xy = 1 Â· y Step 2 Group terms with common factors. (6x2 + 3xy) + (2x + y) = (3x Â· 2x + 3x Â· y) + (1 Â· 2x + 1 Â· 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).