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What is Factoring by Grouping?

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.
ScienceStruck Staff
Last Updated: Jan 18, 2019
A polynomial is an expression containing variables, constants, and exponents (must be non-negative and whole number) which are expressed with the operations of addition, subtraction, and multiplication. Their use in science and economics make them a must-know feature of mathematics.
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:
9a3 - 15a2 + 3ab - 5b

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

9a3 + 3ab - 5b - 15a2
=> 3a (3a2 + b) - 5(3a2 + b) (...taking the common factors out)
=> (3a2 + 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.
★ If the expression is correct, then proceed to check if the terms in the expression contain GCF (Greatest common factor). If there is one, simplify the expression by removing it.
★ Now check if there are any terms that contain common factors. If yes, group them together, and further remove the GCF of each group.

★ 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!

Example # 1:
y2 + 8xy + 2y + 16x
=> y2 + 2y + 8xy + 16x (...grouping the common factors)
=> y (y + 2) + 8x (y + 2) (...applying distributive law)
=> (y + 8x) (y + 2)
Example # 2:
6a3 - 9a2 + 2ab - 3b
=> 6a3 + 2ab - 3b - 9a2 (...grouping the common factors)
=> 2a (3a2 + b) - 3 ( 3a2 + b) (...applying distributive law)
=> (2a - 3) (3a2 + b)
Now let's have a look at trinomial expressions. A trinomial expression, as the name suggests, contains three terms, each containing an exponent in increments.
The expression is written as ax2 + 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: ax2 + 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.
Let's solve some trinomial expressions now!

Example # 3:
2x2 + 13x + 15 (.....a = 2, b = 13, c = 15)
=> 2x2 + 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:
6x2 - 19x + 10 (...a = 6, b = -19, c = 10)
=> 6x2 - 15x - 4x + 10 (...product of a and c = 60, common factor is 15 x 4, as 15 + 4 = 19, which is b)
=> 6x2 - 4x - 15x + 10
=> 2x (3x - 2) - 5 (3x - 2) (...grouping common terms)
=> (2x - 5) (3x - 2) (...Applying distributive law)
Understand the rules thoroughly before you attempt to solve these equations. Because, if the basics are clear, even complex polynomial equations will be a cakewalk for you! Good luck!