Warm - up 6.4Warm - up 6.4Factor:

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 = -5

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

6.4 solving 6.4 solving polynomial polynomial equationsequations

by Jason L. Bradbury

CA State Standard

- 3.0 Students are adept at operations on polynomials, including long division.

- 4.0 Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.

Objective – To be able to factor and solve polynomial expressions.

2x2x22 – 5x – 12 – 5x – 12

In Ch. 5 we learned how to factor:In Ch. 5 we learned how to factor:

- A General Trinomial- A General Trinomial

6.4 solving polynomial 6.4 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 4x4x22 – 9 – 9 (2x)(2x)2 2 – 3– 322

(2x + 3)(2x – 3)(2x + 3)(2x – 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 1

FactorFactor

(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 + 8a = 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 3

FactorFactor

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 2

Factor by groupingFactor by grouping

-9(x – 2)

(x – 3)(x + 3)(x – 2)

6.4 Homework6.4 Homework

Page 336 – 337 Page 336 – 337 12 – 14, 21 – 27, and 3112 – 14, 21 – 27, and 31

6.4 Guided 6.4 Guided PracticePractice

Page 336 – 337 Page 336 – 337 12 – 14 and 21 – 2412 – 14 and 21 – 24