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)

Difference of Squares

The rule is:

A²-B² = (A-B)(A+B)

So

x² -25 = (x+5)(x-5)
4 - 9w² = -9w² + 4 = (2-3w)(2+3w)

Perfect Square Trinomial

The rule is:

A² + 2AB + B² = (A+B)²
A² - 2AB + B² = (A-B)²

So

x² + 6x + 9 = (x-3)²

Sum and Difference of Cubes

The rule is:

A³ + B³ = (A + B)(A² - AB + B²) // Sum
A³ - B³ = (A - B)(A² + AB + B²) // Difference

So

x³ - 64 =  x³ - 4³ = (x-4)(x² + 4x + 16)

Note the sign of the original equation does not change that A = x and B = 4

Factoring by Substitution

So this is where you substitute a term and make it easier to spot how to approach. For example

2x⁴ - 7x² + 3

We can let k = x² and we have

2k² - 7k + 3

We can now use the Factoring a Quadratic Trinomial (a <> 1) Remember ax² + bx + c

2k² - 7k + 3, a = 2, b = -7, c = 3
_ + _ = -7
_ * _ = 6

So -6 and -1 fit the bill

 2k² -1k - 6k  + 3

So using grouping we have

  k(2k -1) - 3(2k -1)
= (2k - 1)(k - 3)

Replace the k with x²

= (2x² - 1)(x² - 3)