Warm - upWarm - upFactor:1. 4x2 – 24x 4x(x – 6)

2. 2x2 + 11x – 21 (2x – 3)(x + 7)

3. 4x2 – 36x + 81 (2x – 9)2

Solve:4. x2 + 10x + 25 = 0

x = -55. 6x2 + x = 15 x = 3/2 and -5/3

2x2x22 – 5x – 12 – 5x – 12 We know how to factor:We know how to factor:- A General Trinomial- A General Trinomial

Solving Polynomial Solving Polynomial EquationsEquations

(2x + 3)(x – 4) (2x + 3)(x – 4) - A Perfect Square Trinomial- A Perfect Square Trinomial xx22 + 10x + 25 + 10x + 25

(x + 5)(x + 5) = (x +5)(x + 5)(x + 5) = (x +5)22

- The Difference of two Squares- The Difference of two Squares xx22 – 9 – 9 (x)(x)2 2 – 3– 322

(x + 3)(x – 3)(x + 3)(x – 3)- A Common Monomial Factor- A Common Monomial Factor 6x6x2 2 + 15x + 15x

3x(2x + 5)3x(2x + 5)

a) x4 – 6x2 – 27

Example 1Example 1FactorFactor

(x2 + ?)(x2 – ?)(x2 + 3)(x2 – 9)(x2 + 3)(x – 3)(x + 3)

b) x4 – 3x2 – 10(x2 + ?)(x2 – ?)(x2 + 2)(x2 – 5)

aa33 + b + b33 = (a + b)(a = (a + b)(a22 - ab + b - ab + b22))

aa33 + b + b33 = (a + b)(a = (a + b)(a22 - ab + b - ab + b22))

Sum of Two CubesSum of Two Cubes ** Special Factoring Patterns** Special Factoring Patterns

ex. xex. x33 + 8 + 8 a = xa = x

(x + 2)(x(x + 2)(x22 – 2x + 4) – 2x + 4)

aa33 – b – b33 = (a – b)(a = (a – b)(a22 + ab + b + ab + b22))

Example 2Example 2xx3 3 + 125 + 125

xx33 + 5 + 53 3

Difference of Two CubesDifference of Two Cubes

b = 2b = 2

ex. 8xex. 8x33 – 1 – 1

xx33 + 2 + 233

a = 2xa = 2x

(2x – 1)(4x(2x – 1)(4x22 + 2x + 1) + 2x + 1) b = 1b = 1

(2x)(2x)33 – (1) – (1)33

= (x + 5)(x= (x + 5)(x22 – 5x + 25) – 5x + 25)

a) x3 – 27

Example 3Example 3FactorFactor

aa33 – b – b33 = (a – b)(a = (a – b)(a22 + ab + b + ab + b22))

xx33 – 3 – 33 3 = (x – 3)(x= (x – 3)(x22 + 3x + 9) + 3x + 9)

b) 8x3 + 64aa33 + b + b33 = (a + b)(a = (a + b)(a22 - ab + b - ab + b22))

(2x)(2x)33 + (4) + (4)3 3 = (2x + 4)(4x= (2x + 4)(4x22 – 8x + 16) – 8x + 16)

Must be the sameMust be the same

x2(x – 2)

x3 – 2x2 – 9x + 18

(x2 – 9)(x – 2)

Extra Example 2Extra Example 2Factor by groupingFactor by grouping

-9(x – 2)

(x – 3)(x + 3)(x – 2)

Solving Polynomial EquationsSolving Polynomial Equations1. Factor out GCF2. Factor remaining

quadratic equation1. If remaining

equation can not be factored, use quadratic formula.

3. Solve all equations for variable.

01223 xxx#1: 0122 xxx

034 xxx0x 04x

4x03x

3x

0189 234 xxx#3:

018922 xxx 0362 xxx02 x 06x

6x03x3x0x

xxx 44 23 #7:24x24x

xxx 44 23 x4x4

044 23 xxx 0442 xxx 022 xxx0x 02x

2x02x

2x

045 24 xx#6:

014 22 xx

22 xx 011 xx02x

2x02x

2x01x1x

01x1x

03 23 xxx#1B:

0132 xxx0x 0132 xx

What do we do when we can’t factor?

131

cba

aacbbx

242

12

11433 2 x

253

x

Roots:

253,

253,0

010142 345 xxx#4B:

0572 23 xxx02 3 x 0572 xx

What do we do when we can’t factor?

571

cba

aacbbx

242

12

51477 2 x

2297

x

Roots:

2297,

2297,0

0x

08208 23 xxx#1M:

02524 2 xxx04 x 0252 2 xx

What do we do when we can’t factor?

252

cba

aacbbx

242

22

22455 2 x

435

495

x

Roots:

0x

248

435

x

21

42

435

x

21,2,0

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