Factoring by Grouping

Math has always been mind-boggling for me. How much ever people take an instant dislike for this rattling subject, its use in different walks of life remains undisputed. And one such interesting aspect of this subject is solving polynomial expressions. But before I start with my article, I would first like to shed some light on what are polynomials? Well, 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 the polynomial expression below:

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. By this statement I mean, 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 contains 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 the 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. So follow the steps below 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 above 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 cake walk for you! Good luck!