Factoring Polynomials by Grouping

Introduction of factoring polynomials by grouping:

• In general, the sum of a countable number of monomials is referred as a polynomials. The way of writing a polynomials as a product of two or more simpler polynomials is calledfactorization. The process of factorization is also known as the resolution into factors or factoring polynomials. Each simpler polynomial in the product is called a factor of the given polynomial.   
• Factoring polynomial by grouping method is used for the expressions containing more than three terms.

Factoring Polynomials Using Grouping:

Factoring polynomials by grouping involving four or more than four terms can be grouped using the following steps:

Step 1:  The terms having common factors should be grouped.

Step 2: The Greatest Common Factor (GCF) is taken out.

Step 3:  In the third step, we have to use the distributive law to find the factors.

                                               The distributive law is given by

                                                         a (b + c) = a b + a c

Example Problems on Factoring Polynomials by Grouping:

Ex 1:  Factoring the given polynomial, 3x² + 9x³ + 7x⁷ + 21x⁸, by grouping:     

Sol :   Step 1:  The terms having common factors should be grouped.

                                     (3x² + 9x³⁾ + (7x⁷ + 21x⁸).

Step 2: The Greatest Common Factor (GCF) is taken out.

                            3x² + 9x³ and 7x⁷ + 21x⁸ both can have a GCF.

               Taking out the GCF outside, we get

                         (3x² + 9x³⁾+ (7x⁷ + 21x⁸) = 3x² (1 + 3x) + 7x⁷ (1 + 3x)

Step 3: Use  distributive law to find the factors.

                Note that there is a common factor, 1 + 3x. So, factor out 1+3x.

                  By distributive law, we get

                                3x² + 9x³ + 7x⁷ + 21x⁸ = (3x² + 7x⁷) (1 + 3x).

Ex  2:   Factoring the given polynomial by grouping:  2x³+6x²−6x−18

Sol :   Step 1: The terms having common factors should be grouped.

                                               (2x³+6x²) + (−6x−18)

Step 2: The Greatest Common Factor (GCF) is taken out.

                                  (2x³+6x²) and  (−6x−18) both have a GCF

               Taking out the GCF outside, we get

                                   (2x³+6x²) + (−6x−18) = 2x²(x+3)-6(x+3)

Step 3:  Use distributive law to find the factors.

              Note that there is a common factor, x+3. So, Factor out (x+3).

               By distributive law, we get

                                             2x³+6x²−6x−18=(x +3) (2x²-6).

Ex 3 :  Factoring the given expression by grouping: 2x² - 3x + 10x - 15.

Sol :  Step 1:  The terms having common factors should be grouped.

                                       (2x² + 10x) + (- 3x – 15)

Step 2: The Greatest Common Factor (GCF) is taken out.

                            2x²+10x) and  (−3x−15) both have a GCF

              Taking out the GCF outside, we get

                              (2x²+10x) + (−3x−15) = 2x(x+5)-3(x+5)

Step 3:  Use distributive law to find the factors.

              Note that there is a common factor, x+5. So, Factor out (x+5).

               By distributive law,we get

                                             2x² - 3x + 10x – 15=(x+5) (2x-3).

These are the examples for the factoring polynomials by grouping.

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