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Alg2 3.4 Notes.notebook October 15, 2012 34 Factoring Polynomials FACTORING REVIEW NEXT 12 SLIDES *Courtesy of Paul Sullivan Factoring Review Why do we factor quadratic equations? Essential Question: Factor Form Standard Form ax 2 + bx + c = 0 (x - p)(x - q) = 0 Examples x 2 + 5x + 6 = 0 (x + 2)(x + 3) = 0 (3x + 4)(x - 5) = 0 3x 2 - 11x - 20 = 0 Which are easier to solve?

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Alg2 3.4 Notes.notebook October 15, 2012

3­4 Factoring Polynomials

FACTORING REVIEW NEXT 12 SLIDES

*Courtesy of Paul Sullivan

Factoring Review

Why do we factor quadratic equations?Essential Question:

Factor FormStandard Formax2 + bx + c = 0 (x - p)(x - q) = 0

Examplesx2 + 5x + 6 = 0 (x + 2)(x + 3) = 0

(3x + 4)(x - 5) = 03x2 - 11x - 20 = 0

Which are easier to solve?

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Alg2 3.4 Notes.notebook October 15, 2012

Learning Targets: We are reviewing how to factor quadratic equations

We will review the 6 different methods for factoring

Method 1: Taking out the common factorMethod 2: Factoring x2 + bx +c by Product and Sum TableMethod 3: Factor by GroupingMethod 4: Factoring ax2 + bx +c by Splitting the Middle TermMethod 5: Special Products - Difference of SquaresMethod 6: Special Product - Perfect Square Quadratic

*Courtesy of Paul Sullivan

Factoring by Taking out the common Factor

7x2(2x2 - 3)Simplify

Factor2 7 x x x x - 3 7 x x

2 7 x x x x - 3 7 x x7 x x (2 x x - 3)

7x2(2x2 - 3)

(Method 1)

14x4 - 21x2

7 x x (2 x x - 3)

2 7 x x x x - 3 7 x x2 7 x x x x - 3 7 x x

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Alg2 3.4 Notes.notebook October 15, 2012

1.) 2x2 + 8

Factor each expression by taking out the common factor

2.) 7x2 - 14 3.) 4x2 + 16x - 4

4.) 5x2 + 10 5.) 25x2 - 20 6.) 4x3 + 20x2 - 24x

Solve each equation by factoring

1.) 9x2 + 18x = 0 2.) 8x2 - 12x = 0 3.) 5x2 = 15x

(Method 2) Factoring x2 + bx + c (Product and Sum Table)

x2 + 12x + 20

x2 + 14x + 48

product

sum product

c b

2.)

sum

1 202 104 5

list offactor pairs

2112 9

winning factor pair

(x + 2)(x + 10)

(press for hidden)

1.)

Put in factored formproduct

c bsum x2 + x -303.)

productc b

sum

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Alg2 3.4 Notes.notebook October 15, 2012

Factoring by Grouping

x3 + 4x2 + 6x + 24

x3 + 4x2 + 6x + 24first group second group

Step-By-Step1.) Break into two groups.2.) Take out common factor from first group.3.) Take out the common factor from the second group.4.) If the remainder is the same for both groups, factor it out to the first "bubble".5.) The two remaining terms form the second "bubble".

x2(x + 4) + 6x +24

x2(x + 4) + 6(x + 4)

(x + 4)(x2 + 6)

(Method 3):

(press for hidden)

Factor By Grouping

2.) 3x3 - x2 - 21x + 7

5.) 6.)

1.) x3 + 2x2 + 3x + 6

3.) b3 + 5b2 - 24b - 20 4.) c3 + c2 - 12c - 12

Solve each equation by factoring

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Alg2 3.4 Notes.notebook October 15, 2012

Factoring ax2 + bx + c by Splitting the Middle Term *British Method

Ex 1: Factor 2x2 + 11x + 5

a cProduct Sum

b

Step by Step1. Make a Product and Sum Table the Product has to be a c. The sum has to be b.

2.) Split the middle term to get four terms.

3.) Factor by grouping

(Method 4)

More Examples Step by Step1. Make a Product and Sum Table the Product has to be a c. The sum has to be b.

2.) Split the middle term to get four terms.

3.) Factor by grouping

a cProduct Sum

b1.) 3x2 - 4x - 7

2.) 6x2 - 19x + 15

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Alg2 3.4 Notes.notebook October 15, 2012

Factoring Special Products

Difference of Squares Pattern (Sum and Difference Pattern)

(a + b)(a - b) a2 - b2

Perfect Square Pattern(a + b)2 = (a + b)(a + b) a2 + 2ab + b2

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

Special Products Patterns (Short Cuts to FOIL)

Sum and Difference Pattern (Difference of Squares)(a - b)(a + b) = a 2 - b2

Perfect Square Pattern (Square of a Binomial)(a + b)2 = a2 + 2ab + b2

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

These are examples of the two kinds of special products

Write each expression in simplest form

1.) (x + 4)(x - 4) 2.) (x + 7)2 3.) (3x -5)2

Factor each difference of squares

1.) m2 - 25 2.) 4p2 - 81 3.) 25 - 49x2

4.) 49v2 - 144 4.) 81v2 + 100

Press here for more practice problems

(Method 5)

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Alg2 3.4 Notes.notebook October 15, 2012

Factor each Perfect Square Trinomial

1.) x2 + 16x + 64 2.) x2 - 10x + 25

3.) 4x2 + 36x + 81 4.) 9x2 - 30x + 25

5.) x2 - 14x + 49 6.) x2 + 20x + 100

7.) 49x2 - 56x + 16 8.) 36x2 + 60x + 25

Press here for more problems

(Method 6):

3­4 Factoring Polynomials

Use the Factor Theorem to determine factors of a polynomial.

Factor the sum and difference of two cubes.

To find real roots of a polynomial.

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Alg2 3.4 Notes.notebook October 15, 2012

I. Factor Theorem

1. Determine whether a linear binomial is a factor:A. (x + 1); (x2 – 3x + 1) B. (x + 2); (3x4 + 6x3 – 5x – 10)

II. Sum & Difference of Two Cubes

2. Factor 125d3 – 8

(5d – 2)(25d2 + 10d + 4)

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Alg2 3.4 Notes.notebook October 15, 2012

III. All together now...3. Factor (grouping):

4. Factor (cubic):

5. Factor (Quadratic):

x3 – x2 – 25x + 25.

4x4 + 108x

4. 4x(x + 3)(x2 – 3x + 9)

3. (x – 1)(x – 5)(x + 5)

5. (x2 + 9)(x2 ­ 5)

x4 + 4x2 ­ 45

IV. More Fun6. (Homework #28) Factor 24n2 + 3n5

Answers: 6. 3n2(2 + n)(4 ­ 2n + n2)7. x4 ­ 14x2 ­ 32

7. x4 ­ 14x2 ­ 32

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Alg2 3.4 Notes.notebook October 15, 2012

Say I knew (x ­ 7) was a factor of x3 + 3x2 + 2x ­ 504.

How can a write the polynomial as a product?

3­4 p.177 #2, 3, 5, 7, 11, 15, 17, 19, 25, 27, 31, 36, 38,

39, 41, 42, 51, 53, 54, 56, 57

Check: SA11