# Back to Basics: What is Factoring By Grouping?

Factoring by grouping is an efficient way of solving polynomials. A good way to learn about this is through some examples, which this post gives you.

Narayani Karthik

Last Updated: Jan 18, 2019

What is Factoring by Grouping?

Factoring could be defined as grouping terms given in a polynomial expression, which have common factors. For instance, take a look at this polynomial expression:

^{3}- 15a

^{2}+ 3ab - 5b

So, after grouping, the expression would be something like this:

9a

^{3}+ 3ab - 5b - 15a

^{2}

=> 3a (3a

^{2}+ b) - 5(3a

^{2}+ b) (...taking the common factors out)

=> (3a

^{2}+ b) (3a - 5)

Remember, factoring is applicable to expressions containing at least 4 terms, with absolutely nothing in common.

Steps Involved

Here are some steps that will help you understand the technique better:

★ First check if your polynomial expression is correct or not. This mean that the exponents in the expression should not be fractional or negative.

★ First check if your polynomial expression is correct or not. This mean that the exponents in the expression should not be fractional or negative.

★ Apply the distributive law (ab + ac = a (b+c)), and get the simplified expression.

Examples

Now, that the factoring by group steps are clear, let's solve some examples!

y

=> y

=> y (y + 2) + 8x (y + 2) (...applying distributive law)

=> (y + 8x) (y + 2)

*Example # 1*:y

^{2}+ 8xy + 2y + 16x=> y

^{2}+ 2y + 8xy + 16x (...grouping the common factors)=> y (y + 2) + 8x (y + 2) (...applying distributive law)

=> (y + 8x) (y + 2)

*Example # 2*:

6a

^{3}- 9a

^{2}+ 2ab - 3b

=> 6a

^{3}+ 2ab - 3b - 9a

^{2}(...grouping the common factors)

=> 2a (3a

^{2}+ b) - 3 ( 3a

^{2}+ b) (...applying distributive law)

=> (2a - 3) (3a

^{2}+ b)

^{2}+ bx + c, where the power of x is reduced to 0 for a constant. The technique used here in factoring is almost the same, with some subtle changes. Here are the steps to solve trinomial expressions by factoring method.

- Identify the constants in the expression: ax
^{2}+ bx + c - Multiply the constants, a and c.
- Now try to find 2 positive factors of the product, whose sum equals b.
- Once calculated, expand the expression and rewrite it.
- Now use the earlier mentioned steps of factoring by grouping, to get the simplified expression.

*Example # 3*:

2x

^{2}+ 13x + 15 (.....a = 2, b = 13, c = 15)

=> 2x

^{2}+ 10x + 3x + 15 (...product of a and c = 30, common factors is 10 x 3, as 10 + 3 = 13, which is b)

=> 2x (x + 5) + 3 (x + 5) (...grouping common terms)

=> (2x + 3) (x + 5) (...applying distributive law)

*Example # 4*:

6x

^{2}- 19x + 10 (...a = 6, b = -19, c = 10)

=> 6x

^{2}- 15x - 4x + 10 (...product of a and c = 60, common factor is 15 x 4, as 15 + 4 = 19, which is b)

=> 6x

^{2}- 4x - 15x + 10

=> 2x (3x - 2) - 5 (3x - 2) (...grouping common terms)

=> (2x - 5) (3x - 2) (...Applying distributive law)