Advanced Algebra / Trigonometry

Section 6-5Solving Polynomial Equations

(Factoring Review)

Target Goals

1) Factor by taking out the greatest common factor.2) Factor by grouping.3) Factor trinomials.4) Factor the difference of two squares.5) Factor the sum and difference of two cubes.

Please select a Team.

A. B. C. D. E. F. G. H. I. J.

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A. Team 1B. Team 2C. Team 3D. Team 4E. Team 5F. Team 6G. Team 7H. Team 8I. Team 9J. Team 10

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FactoringVocabFactoring: To factor a polymonial means to express the polynomial as the

product of its prime factors.** Factoring is the Inverse/Opposite of distributing.** Factoring is the Inverse/Opposite of FOILing

1:Ex 6 8 Distr2 3 4 ibutingxx 2 3 4 Factorin8 g6 xx

2 2 15 FOILing/Di3 5 stributingx x x x 2 3 5 Factor g15 in2 x xx x

2 :Ex

210 6 15 9 FOILing/Distributi5 g3 2 3 nx y x x xy x y 210 6 15 9 5 3 2 3 Factoringxx x yy x y x

3 :Ex

2 3 8 D2 i2 stributing4x x x x 23 2 2 4 Factori g8 nx x xx

4 :Ex

Factoring by taking out the greatest common factor

Example 1Factor.

4 230 15a a 2 215 2 1a a

Example 22 2 216 24 40xy y z y 28 2 3 5y x z

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a. b. c. d.

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Target Goal: Factor by taking out the greatest common factor.

.C

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2 216 12m n mn

4 12 8mn m n

4 4 3mn m n

2 8 6mn m n

2 24 4 3m n n m

Factor.

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Target Goal: Factor by taking out the greatest common factor.

.C

.D

3 2 2 3 436 66 210x y x y xy

2 2 26 6 11 35xy x xy y

2 2 26 6 11 35xy x y xy y

3 4 2 26 6 11 35x y y xy x

2 2 22 18 33 105xy x xy y

Factor.

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Factoring by Grouping

Example 3Factor.

22 2 10 10abc abd cd d 22 2 10 10abc abd cd d

Example 43 25 5x y x x y

3 25 5x y x x y

Use this method when you have 4 or more terms.

Steps used to factor by grouping.1. "Group" the polynomial into smaller polynomials using parentheses. 2. Factor out the greatest common factor from each "group". 3. The remaining factor in each group should be the same. Factor this out.4. If the remaining factor in each group in not the same, regroup and try again.

2 10ab c d d c d

2 10c d ab d

3 5 5x y x xy Remaining factors not the same

3 25 5x x y x y

2 25 5x x y x 2 5x x y

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a. b. c. d.

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Target Goal: Factoring by Grouping.

.C

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6 8 9 12xy x y

3 4 2 3y x

3 4 2 3y x

3 4 2 3y x

3 4 2 3y x

Factor.

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Target Goal: Factoring by Grouping.

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2 7 7a ab b a

7a b a

Cannot be factored

7a b a

7a b a

Factor.

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Factoring Trinomials

Example 5Factor.

2 8 15x x

Example 6216 24 9r r

3 2 5x x

Came from the "F" in FOIL.

23 15 2 10x x x 23 13 10x x

Came from the "OI" in FOIL.

Came from the "L" in FOIL.

Putting the "Puzzle" together

x x5533

15 11 15

16r

2rr

8r4r 4r

3 31

Multiply for this sign.Add for this sign.

3 5 x x

919

3 4 3 4r r

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a. b. c. d.

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Target Goal: Factoring Trinomials.

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24 7 3x x

2 3 2 1x x

2 1 2 3x x

4 3 1x x

4 3 1x x

Factor.

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Target Goal: Factoring Trinomials.

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22 5 12x x

2 2 6x x

2 3 4x x

2 4 3x x

2 6 2x x

Factor.

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Factoring the Difference of Two Squares

Example 72 25x

Perfect SquaresNumbers Variables

1

81644936251694

196169144121100

225

2x

10x8x6x4x Difference of Perfect Squares

2 2a b a b a b

5 5x x

Example 82 21 4

9 25r s

1 2 1 23 5 3 5r s r s

Example 94 416x z

2 2 2 24 4x z x z

2 24 2 2x z x z x z

Factoring Sums and Differences of Two Cubes

Example 103 125x

Perfect CubesNumbers Variables

1

72951234321612564278

1000

3x

15x12x9x6x

Sum of Perfect Cubes 3 3 2 2a b a b a ab b

25 5 25x x x

Difference of Perfect Cubes 3 3 2 2a b a b a ab b

Same Opposite PositiveAlways

SOAP

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a. b. c. d.

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a. b. c. d.

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Target Goal: Factoring Difference of Perfect Squares.

.C

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2 49a b

2 23 3a b a b

23 3 3a b a b a b

2 29a b a b

3 3a b a b

Factor.

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a. b. c. d.

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Target Goal: Factoring Sum or Difference of Perfect Cubes.

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3 64x

24 4 16x x x

2 8 8x x

4 4 4x x x

24 4 16x x x

Factor.

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Summary of Factoring

Example 11

#1 Take out common factors.

25 45ax a 4 2 312x x x 25 9a x

Checklist for Factoring Polynomials

#2 If binomial, look for difference of perfect squares. If binomial, look for sum and difference of perfect cubes. If trinomial, "put the puzzle together." If 4 or more terms, factor by grouping.#3 If polynomial doesn't factor, it is "prime".

5 3 3a x x

Example 12

2 2 12x x x

2 2 12x x x

2 4 3x x x

Example 1338 27x

22 3 4 6 9x x x

Example 143 23 4 12 16x x x

3 23 4 12 16x x x 2 3 4 4 3 4x x x 23 4 4x x

3 4 2 2x x x

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Target Goal: Factoring.

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29 58 49y y

9 49 1y y

3 7 3 7y y

3 7 3 7y y

9 7 7y y

Factor.

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Target Goal: Factoring.

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23 12s

3 4 4s s

23 4s

3 2 2s s

23 12s

Factor.

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Target Goal: Factoring.

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29 9 2 2a am a m

29 2a am a m

9 2a m a

9 2a m a

Factor.

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Prime

Team Scores166.67 Team 7133.33 Team 8100 Team 5100 Team 466.67 Team 350 Team 250 Team 1050 Team 650 Team 940 Team 1

End

Advanced Algebra / Trigonometry

Section 6-5Solving Polynomial Equations

(Factoring Review)

Target Goals

1) Factor by taking out the greatest common factor.2) Factor by grouping.3) Factor trinomials.4) Factor the difference of two squares.5) Factor the sum and difference of two cubes.

FactoringVocabFactoring: To factor a polymonial means to express the polynomial as the

product of its prime factors.** Factoring is the Inverse/Opposite of distributing.** Factoring is the Inverse/Opposite of FOILing

1:Ex 6 8 Distr2 3 4 ibutingxx 2 3 4 Factorin8 g6 xx

2 2 15 FOILing/Di3 5 stributingx x x x 2 3 5 Factor g15 in2 x xx x

2 :Ex

210 6 15 9 FOILing/Distributi5 g3 2 3 nx y x x xy x y 210 6 15 9 5 3 2 3 Factoringxx x yy x y x

3 :Ex

2 3 8 D2 i2 stributing4x x x x 23 2 2 4 Factori g8 nx x xx

4 :Ex

Factoring by taking out the greatest common factor

Example 1Factor.

4 230 15a aExample 2

2 2 216 24 40xy y z y

Factoring by Grouping

Example 3Factor.

22 2 10 10abc abd cd d Example 4

3 25 5x y x x y

Use this method when you have 4 or more terms.

Steps used to factor by grouping.1. "Group" the polynomial into smaller polynomials using parentheses. 2. Factor out the greatest common factor from each "group". 3. The remaining factor in each group should be the same. Factor this out.4. If the remaining factor in each group in not the same, regroup and try again.

Factoring Trinomials

Example 5Factor.

2 8 15x x Example 6

216 24 9r r

3 2 5x x 23 15 2 10x x x

23 13 10x x

Factoring the Difference of Two Squares

Example 72 25x

Perfect SquaresNumbers Variables

1

81644936251694

196169144121100

225

2x

10x8x6x4x Difference of Perfect Squares

2 2a b a b a b

Example 82 21 4

9 25r s Example 9

4 416x z

Factoring Sums and Differences of Two Cubes

Example 103 125x

Perfect CubesNumbers Variables

1

72951234321612564278

1000

3x

15x12x9x6x

Sum of Perfect Cubes 3 3 2 2a b a b a ab b

Difference of Perfect Cubes 3 3 2 2a b a b a ab b

Summary of Factoring

Example 11

#1 Take out common factors.

25 45ax a 4 2 312x x x

Checklist for Factoring Polynomials

#2 If binomial, look for difference of perfect squares. If binomial, look for sum and difference of perfect cubes. If trinomial, "put the puzzle together." If 4 or more terms, factor by grouping.#3 If polynomial doesn't factor, it is "prime".

Example 12 Example 1338 27x

Example 143 23 4 12 16x x x