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Narayani Karthik
May 13, 2019

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.

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:

9a^{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.

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

9a

=> 3a (3a

=> (3a

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.

★ 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.

★ 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*:

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)

y

=> y

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

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

6a

=> 6a

=> 2a (3a

=> (2a - 3) (3a

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 ax^{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.

Let's solve some trinomial expressions now!

*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)

2x

=> 2x

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

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

6x

=> 6x

=> 6x

=> 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!