Factoring

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=

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)