What is Factoring by Grouping?

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.

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.

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.

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!