2.3 Part 1Factoring

10/29/2012

What is Factoring?

It is finding two or more numbers or algebraic expressions, that when multiplied together produce a given product.Ex. Factor 6: 2 • 3 Factor 2x2 + 4: 2 • (x2 +2) Factor x2 +5x + 6: (x+2)•(x+3)

Type 1 Problems

Factoring Quadratic equations in Standard form

y = ax2 + bx +cwhen a = 1

and when a > 1

The Big “X” method

c

b

Think of 2 numbers that Multiply to c and Add to b

#1 #2

add

multiply

Answer: (x ± #1) (x ± #2)

Factor: x2 + bx + cNote: a = 1

15

8

Think of 2 numbers that Multiply to 15 and Add to 8

3 x 5 = 153 + 5 = 8

3 5

Answer: (x + 3) (x + 5)

Factor: x2 + 8x + 15

c

b

#1 #2

add

multiply

8

-6

Think of 2 numbers that Multiply to 8 and Add to -6

-4 x -2 = 8-4 + -2 = -6-4 -2

Answer: (x - 4) (x - 2)To check: Foil (x – 4)(x – 2) and see if you get x2-6x+8

Factor: x2 - 6x + 8

c

b

#1 #2

add

multiply

-9

8

Think of 2 numbers that Multiply to -9 and Add to 8

9 x -1 = -99 + -1 = 8

-9 8

Answer: (x - 9) (x + 8)

Factor: x2 + 8x - 9

c

b

#1 #2

add

multiply

The Big “X” method

a•c

b

Think of 2 numbers that Multiply to a•c and Add to b

#1 #2

add

multiply

Answer: Write the simplified answers in the 2 ( ) as binomials. Top # is coefficient of x and bottom # is the 2nd term

Factor: ax2 + bx + cNote: a > 1

a aSimplify like a fraction if needed

Simplify like a fraction if needed

3•2 = 6

7

Think of 2 numbers that Multiply to 6 and Add to 7

6 x 1 = 66 + 1 = 76 1

Answer: (x + 2) (3x + 1)

Factor: 3x2 + 7x + 2

a•c

b

#1 #2

add

multiply

3 3Simplify like a fraction . ÷ by 3

2

1

a a

4(-9) = -36

-16

Think of 2 numbers that Multiply to -36 and Add to -16

-18 x 2 = -36 -18 + 2 = -16

-18 2

Answer: (2x - 9) (2x + 1)

Factor: 4x2 - 16x - 9

a•c

b

#1 #2

add

multiply

4 4Simplify like a fraction . ÷ by 2

-9

2

a a1

2 Simplify like a fraction . ÷ by 2

Type 2 ProblemsFactoring Quadratic equations

written as Difference of 2 Squares.

Difference of Two Squares Pattern

(a + b) (a – b) = a2 – b2

In reverse, a2 – b2 gives you (a + b) (a – b)Examples: 1. x2 – 4 = x2 – 22 = (x + 2) (x – 2)

2. x2 – 144 =(x + 12) (x – 12)3. 4x2 – 25 = (2x + 5) (2x – 5)

If you can’t remember that, you can still use the big X method.

Factor: x2 – 4

-4

0

Think of 2 numbers that Multiply to -4 and Add to 0

2 x -2 = -42 + -2 = 0

2 -2

Answer: (x + 2) (x - 2)

Ex. x2 + 0x – 4

Ex. x2 – 144

-144

0

Think of 2 numbers that Multiply to -144 and Add to 0

12 x -12 = -14412 + -12 = 0

12 -12

Answer: (x + 12) (x - 12)

x2 + 0x – 144

4(-25) = -100

0

Think of 2 numbers that Multiply to -100 and Add to 0

-10 x 10 = -100 -10 + 10 = 0

-10 10

Answer: (2x - 5) (2x + 5)

Factor: 4x2 - 25

4 4Simplify like a fraction . ÷ by 2

-5

2

5

2 Simplify like a fraction . ÷ by 2

4x2 + 0x - 25

Type 3 ProblemsFactoring Quadratic equations by taking out the Greatest Common

Factor

Factor y = x2 – 6x

1. Find the GCF.GCF = x2. Factor the GCF out. Think reverse “distributive prop.”y = x (x – 6)

Factor y = -8x2 + 18 1. Find the GCF.GCF = -2Why -2 and not 2 you ask? Wait for the next step.

2. Factor the GCF out. y = -2 (x2 - 9)Answer: So we can have the difference of 2 squares pattern

y = 2 (-x2 + 9) Not Difference of 2 Squares3. Factor what’s in the ( ) since it follows the difference of 2 square pattern. y = -2(x – 3)(x + 3)