Warm UpFactor the following.

1) 20x2 - 115x – 30 2) x2 + 4x – 96

3) 14a2 b - 63a5 b6 4) 12x3 +3x2 +20x +5

Factoring-Special Cases

September 4th

So what is a special case? • Multiply (x – 2) (x + 2)….

• This product is a little different than the rest. What is it missing?

• A middle term!

Using what you know…• If given x2 – 4, and asked to factor,

how could you set this up using what you know already?

• What is the middle coefficient, b ?• What is the last number, c ?

• Can you find two numbers that add to be zero and multiply to be – 4 ?

-4

0

Difference of Two Squares

a2 - b2 = (a - b) (a + b)x2 - 22 = (x - 2) (x + 2)x2 – 4 = (x - 2) (x + 2)

This only works for the

DIFFERENCE, not sum/addition!

Example 1Factor x2 - 9 = (a - b) (a + b)

What number squared is 9?

So… (x - 3) (x + 3)

Check your answer by FOIL or box!

Example 2• What if there is a coefficient in the

front?4x2 – 25

It works the same way! What number squared is 4? 25?

(2x - 5) (2x + 5)

You try!1) x2 – 144

2) w2 – 64

3) 16m2 – 49

4) x2 + 25

Another special case…• Multiply (x + 6) (x + 6)….

• What do you notice about the product? Can you find a pattern?

Perfect-Square Trinomials: +

a2 + 2ab + b2 = (a + b) (a + b)

x2 + 8x + 16 = (x + 4) (x + 4)x2 +2(1)(4) + 42 = (x + 4) (x + 4)

If you are having trouble recognizing the pattern,

practice factoring like we did earlier.

Examples1) x2 + 6x + 9

2) x2 + 10x + 25

Perfect-Square Trinomials: -

a2 - 2ab + b2 = (a - b) (a - b) x2 - 14x + 49= (x - 7) (x - 7)x2 – 2(1)(7) + 72 = (x - 7) (x - 7)

Why do we ADD b2?

Examples1) x2 - 10x + 25

2) x2 - 20x + 100

Example • What if there is a coefficient in the

front?4x2 – 12x + 9

What number squared is 4? 9? (2x - 3) (2x - 3)

Why is there a 12x in the middle? Check your answer!

Examples1) 4x2 + 36x + 81

2) 25z2 + 40z + 16

You try!1) 9n2 – 42n + 49

2) 36d2 – 60d + 25

Example• Is 24g2 -6 a difference of two

squares?• What should I do first? • GCF =

• So…. 24g2 – 6 = 6 (4g2 – 1) = 6 (2g - 1) (2g

+ 1)Now factor using difference of squares!

You try!1)27x2 + 90x + 75

2) 8z2 - 64z + 128

Example• Find the side length of the square!

Area = 25r2 - 30r + 9

Factor Completely!!1. 18p2 – 162

2. 50m3 – 98m

Sum of Cubes• a3 + b3 = (a + b)(a2 – ab + b2)

• 27x3 + 1 = (3x)3 + (1)3

= (3x + 1)((3x)2 – (3x)(1) + 12)

= (3x + 1)(9x2 – 3x + 1)

ExamplesExample 1: 8k3 + 729

Example 2: 343u6 + 216

Difference of Cubes• a3 – b3 = (a – b)(a2 + ab + b2)

x3 – 8 = (x)3 - (2)3

= (x – 2) (x2 + (2)(x) + 22) = (x – 2) (x2 + 2x + 4)

ExamplesExample 1: 64x3 – 27

Example 2: 125p3 - 512

So in general…• a3 ± b3 =

(a [same sign] b)(a2 [opposite sign] ab [always positive] b2)

Challenge Question #1Factor: c10 – 30c5d + 225d2

Challenge Question #2If 49x2 – kx + 36 is a perfect square

trinomial, what is the value of k?

Homework• Worksheet

• Start thinking about your test...it’s on TUESDAY!!!