All Factoring Methods

Introduction

So the video promise the ten most common factoring methods. But before all that. What is factoring?

Factoring is the process of breaking down an expression into a product of simpler expression (factors). So

x² + 5x + 6  = (x"+2)(x+3)

Common Factoring

The process of factoring out the GCF (greatest common factor) from ALL terms in the expression

4x + 8 = 4(x+2)

Factoring by Grouping

This is where you

• group the terms
• factor each group
• factor out the common polynomial

  2x² + 6x + 3x + 9 
= 2x(x + 3) + 3(x + 3) Note (x+3) is common and is the GCF
= (x + 3) (2x + 3)

Factoring a Quadratic Trinomial where (a = 1)

Quadratics take the form

ax² + bx + c

If a = 1 then it can be written

x² + bx + c

So for this we need to know the rule for quadratics (similar to difference of two squares - but different)

x² + bx + c = (x+m)(x+n)
where m+n = b and m*n = c

For example

x² + 10x + 24 

Factors of 24 are 1,24, 2,12, 3,9, 4,6 - so 4+6 = 10

(x+4)(x+6) 

For example 2

x² + 4xy - 21y²

Factors of -21 are -1,21, -3,7 - so 7-3 = 4

(x-7y)(x-3y) 

Factoring a Quadratic Trinomial (a <> 1)

Quadratics take the form

ax² + bx + c

There a three steps to do this

• Check for GCF
• Replace bx with two terms whose coefficients have a sum of 'b' and a product of a*c
• Factor by grouping

Example 1

3x²-11x-4 so coefficients are a = 3, b = -11 c = -4

Following the rule

_ + _ = -11
_ * _ = 3(-4)

or

_ + _ = -11
_ * _ = -12

And the answer is -12, +1 so our equation becomes

3x² -12x + 1x - 4

We can now factor my grouping

3x(x-4) + (x - 4)

Example 2

4x²-6x-40

So taking out the GCF

2(2x² -3x -20) so coefficients are a = 2, b = -3 c = -20

Following the rule

_ + _ = -3
_ * _ = 2(-20) = -40

And the answer is -8, 5 so our equation becomes

2(2x² -8x -5x  -20)
2[2x(x+4) - 5(x+4)]
2(x+4)(2 x+5)