Secondary II Math

Utah Integrated Mathematics CoreStudent Edition - Honors

Unit 4: Factoring

Cache County School District 2013-2014

Secondary II Unit 4 – Factoring and Solving Quadratics by Factoring: Table of Contents

Homework Help (QR Codes and links to videos, tutorials, examples)………………….Section 4.1 – Greatest Common Factor, Teacher Notes

Notes, Assignment Section 4.2 – Factoring by Grouping, Teacher Notes

Notes, Assignment Section 4.3 – Difference of Square and Perfect Square Trinomials Task Teacher Notes, Notes, Assignment Section 4.4 – Factoring when a=1 Task, Teacher Notes

Notes, AssignmentFactoring Matching Activity…………………………………………………………… Section 4.1-4.4 Review Worksheet ……………………………………………………….Section 4.5 – Factoring when a does not equal 1, Teacher Notes

Notes, Assignment Section 4.6 –Additional Factoring, Teacher Notes

Notes, Assignment Section 4.7 – Factoring Review Activity, Review Assignment ……………………………....Section 4.8 – Factoring Simple Quadratic Expressions over the Complex Number System Task, Teacher Notes, Notes, Assignment………………………….……………………… Factoring Review Puzzle …………………..…………………………………Review Worksheets

#1 – G.C.F #2 – Difference of Squares #3 – Factoring Trinomials 1 #4 – Factoring Binomials and Trinomials #5 - Reviewing Factoring Skills #6 – Factoring Polynomials Completely

Secondary II Unit 4: Factoring and Solving Quadratics by Factoring Homework Help

Section 4.1

http://goo.gl/Y6azf http://goo.gl/UAFAV

video

http://goo.gl/I9fJn

Section 4.2

http://goo.gl/FBr3H http://goo.gl/f2MMb http://goo.gl/A8QMy

Section 4.3

http://goo.gl/sogMf

video

http://goo.gl/HTmYP

Difference of squares

http://goo.gl/qEaWs

Section 4.4

http://goo.gl/GdDfw http://goo.gl/IWngu http://goo.gl/pjUoT

Section 4.5

http://goo.gl/Ma7Y0http://goo.gl/Xn4mI http://goo.gl/Pa8z0

Section 4.6No additional resources for this section. Use resources above. And www.cachemath2.wordpress.com

Section 4.7No additional resources for this section. This section is a review. Use resources above. And www.cachemath2.wordpress.com

Section 4.7No additional resources for this section. This section is a review. Use resources above. And www.cachemath2.wordpress.com

Unit 4 Lesson 1 – Greatest Common FactorTask 4.1

Name__________________________________Date_________ Hour_______

Complete the task following these simple rules:

1. You can only use multiplication. 2. You can only use the numbers 2, 3, 5, and 7 and they may be repeated.

40=¿ _________∙_________∙ __________∙_________

36 = _________∙_________∙ __________∙_________

48 =

72 =

128 =

135 =

675 =

112 =

210 =

189000 =

What is special about the numbers I had you use?

What does it mean to be a “multiple” of a number?

What does it mean to be a “factor” of a number?

What other methods can I use to find these prime factorizations without guessing?

Let’s try a few examples using these methods:

Unit 4 Lesson 1 – Greatest Common FactorNotes 4.1

Note that:21 x4=3 ∙ 7 ∙ x ∙ x ∙ x ∙ x - and - 18 x2=2∙ 3 ∙3 ∙ x ∙ x and so on.

Using prime factorizations, find the GCF between the following sets of numbers. Example 1: a. 40∧36 b. 56∧70

c. 21 x4∧18 x2 d. 70 x3∧42 x2

e. 60 ab∧126 a2 b2 f. 42 x3∧90 x

g. 6 x3 y∧20 x y2 h. 4 x3 y2∧18 x2 y

Example 2: a. 12 ,28 ,36 b. 63 ,81 ,18

c. 21 , 35 ,63 d. 25 ,50 ,60

Example 3: Find the GCF of the following terms. Then, factor out the GCF and rewrite an equivalent polynomial expression.

a. 6 x2+9 x b. 12 x4+21 x2−15 x

c. 2 x2+6 x+8 d. 8w4−3 w3+5 w2

e. 9 z3−3 z2+15 z f. 4 x2−12 x−16

g. 5 x5+10 x4+15 x3 h. −8 x4−32 x3+16 x2

Unit 4 Lesson 1 – Greatest Common FactorsReady, Set, Go! - Assignment 4.1

http://goo.gl/Y6azf

Name______________________________

Date_________ Hour_______

Ready1. True or False. If the statement is false, then give the correct statement.

a. There are only nine prime numbers. _____________________

b. The prime factorization of 32 is23 ∙3 _______________________

c. The integer 51 is a prime number. ________________________

d. The GCF for the integers 12 and 16 is 4. ______________________

e. The GCF for the integers 10 and 21 is 1. ______________________

f. The GCF for the polynomial 2 x2−6 x y2is x4 y3 ____________________________

g. For the polynomial 2 x2 y−6 x y2 you could factor out either 2 xy∨−2 xy . _______________

h. The greatest common factor for the polynomial 8 a3b−12a2 bis 4ab .____________________

i. x−7=7−x for any real number x. ____________________

j. −3 x2+6 x=−3 x ( x−2 ) for any real number x. ________________________

2. Find the greatest common factor (GCF) for each group of integers or monomials.

a. 40 , 48 ,88 b. 76 ,84 ,100

c. 66 a3 , 72 a2 b ,120 a4 b3 d. 81 x2 y3 z , 200 x3 y2 z , 539 x4 y4 z

3. Complete the factorization of each monomial. (These are not prime factorizations)

a. 27 x=9 () b. 51 y=3 y ()

c. 24 t 2=8 t ( ) d. 18 u2=3u( )

e. 36 y5=4 y2( ) f. 42 z4=3 z2()

g. −14 m4 n3=2 m4() h. −96 a3b4 c5=−12 ab3 c3( )

SetFactor out the GCF in each expression. Then, factor out the GCF and use it to write the polynomial in an equivalent form.

4. 2 w+4 t 5. 12 x−18 y

6. 24 a−36b 7. x3−6 x

8. 5 ax+5 ay 9. h5+h3

10. −6h5 y2+3h3 y6 11. 2 x3−6 x2+8 x

12. 6 x3+18 x2+24 x 13. 12 x4t +30 x3 t−24 x2 t2

14. 15 x2−9 x y2+6 x2 y

Go!First factor out the GCF and rewrite the expression in an equivalent form; and then factor out the opposite of the GCF and rewrite the expression in an equivalent form..

15. 8 x−8 y

16. −5 x2+10 x

17. a−6

18. 4−7 a

19. −30 b4+75 b3

20. −2 x3+6 x2−2 x

21. 12 u5 v6+18u2 v3−15u4 v5

22. −x+5

Unit 4 Lesson 2 – Factor by GroupingNotes 4.2

Factor out the GCF in each expression and then write an equivalent form of the equation. 1. ( x−3 )a+ (x−3 ) b 2. ( y+4 ) 3+( y+4 ) z

3. x (x−1 )−5 ( x−1 ) 4. a (a+1 )−3 (a+1 )

5. m (m+9 )+(m+9 ) 6. w ( w+2 )2+8 (w+2 )2

Use grouping to write the polynomials in an equivalent form by factor each polynomial completely. Recall that you must factor out the GCF first if possible. 7. bx+by+cx+cy 8. x3+ x2−x−1

9. 12 x3+2 x2−30 x−5 10. 21 k3−84 k2+15 k−60

11. xa+ay+3 y+3 x 12. abc−3+c−3 ab

Additional Notes/Examples:

Unit 4 Lesson 2 – Factor by GroupingReady, Set, Go! - Assignment 4.2

http://goo.gl/FBr3H

Name______________________________

Date_________ Hour_______

ReadyUse grouping to write the polynomials in an equivalent form by factor each polynomial completely. 1. xy+2 y+3 x+6 2. ax+3 y−3 x−ay

3. 4.

5. 6.

7. 8.

9. 10.

SetFactor each expression completely, (by grouping).11. 12 a3−9 a2+4 a−3 12. 2 p3+5 p2+6 p+15

13. 12 n3+4 n2+3 n+1 14. 5 n3−10 n2+3n−6

15. 3 n3−4n2+9 n−12 16. m3−m2+2m−2

Go!Factor each expression completely17. 40 xy+30 x−100 y−75 18. 140 ab−60 a2+168 b−72a

19. 90au−36 av−150 yu+60 yv 20. 16 x2c+8 xyd−16 x2 d−8 xyc

Unit 4 Lesson 3 – Difference of Squares and Perfect Square TrinomialsTask 4.3

Name______________________________

Date_________ Hour_______

Work with your partner to complete the following table and answer the questions bellow:Factoring Difference of Squares

Factors Find the product – show work Final Product

(x + 2)(x – 2)(x + 3)(x – 3)

(2x + 1)(2x – 1)(3x – 2 )(3x + 2)

Compare the factors in the first column and write at least two things they have in common (look for patterns):

*

*

Compare the products in the third column and write at least two things they have in common (look for patterns):

*

*

Using your observations, write a formula for the Difference of Two Squares

a2−b2=¿

Using what you have learned about difference of two squares, factor the following.

1. x2−49=¿ 2. m2−16=¿

3. 4 n2−100=¿ 4. 25 y2−64=¿

Factoring a Perfect Square Trinomial

Factors Find the product – show work Final Product

(x+2)2

(x+3)2

(2 x+1)2

(3 x+2)2

Compare the products in the first column and list at least two things they have in common (look for patterns):

*

*

Compare the factors in the third column and list as least two things they have in common (look for patterns):

*

*

Using your observations, write a formula for the square of a binomial or a perfect square with a sum

(a+b )2=¿

Using what you have learned about perfect squares or squared binomials, find the following products without using the distributive property and writing down your work.

1. (x+2)2=¿ 2. (x+5)2=¿

3. (2 x+3)2=¿ 4.(3 x+1)2=¿

Multiply one squared binomial and find the product to complete the table

Factors (Squared Binomial)

Find the product – show work Final Product

(x−2)2

(x−6)2

(2 x−3)2

(3 x−2 y)2

Write 2 observation or patterns that you see regarding the factors and/or their product.*

*

Using your observations, write a formula for the square of a binomial or a perfect square with a difference

( x− y )2=¿

Using what you have learned about perfect squares or squared binomials, find the following products without using the distributive property:

1. (x−2)2=¿ 2. (x−5)2=¿

3. (2 x−3)2 =

A trinomial is a perfect square trinomial if…

1. the first and last terms are of the form a2∧b2 ( perfect squares )

2. the middle term is 2 ab∨−2ab . (2 ∙ first term ∙ last term¿

Unit 4 Lesson 3 – Difference of Squares and Perfect Square Trinomials

Ready, Set, Go! - Assignment 4.3

http://goo.gl/sogMf

Name______________________________

Date_________ Hour_______

Ready1. True or False. If false, explain why.

a. The polynomial x2+16 is a difference of two squares.

b. The polynomial x2−8 x+16 is a perfect square trinomial.

c. The polynomial 9 x2+21 x+49 is a perfect square trinomial.

d. ( 4 x2+4 )=(2 x+2 )2 for any real number x.

e. The polynomial 16 y+1 is a prime polynomial.

f. The polynomial x2+9 can be factored as ( x+3 ) ( x+3 ) .

g. The polynomial 4 x2−4is factored COMPLETELY as 4 ( x2−1 ) .

h. ( y2−2 y+1 )=( y−1 )2 for any real number y.

i. 2 x2−18=2 ( x−3 ) ( x+3 ) for any real number x.

SetDetermine whether each polynomial can be written as a difference of two squares, a perfect square trinomial, or neither of these.

2. x2−20 x+100 3. x2−10 x−25

4. y2−40 5. a2−49

6. 4 y2+12 y+9 7. 9 a2−30 a−25

8. x2−8 x+64 9. x2+4 x+4

10. 9 y2−25 c2 11. 9 x2+4

12. 9 a2+6 ab+b2 13. 4 x2−4 xy+ y2

Go!

Write an equivalent form of each polynomial by factoring each polynomial completely.

14. a2−144 15. 4 x2−9

16. 1−49c2 17. 100 k2−49

18. f 2−36 19. 20 q2−5 r2

20. 2 x2−8

21. x2+2 x+1 22. y2+4 y+4

23. w2+10 w+25 24. b2−6 b+9

25. 25 y2−10 y+1 26. 9 y2−12 y+4

27. 144 x2+24 x+1 28. x2−2 xy+ y2

29. 9 w2+42 w+49 30. 4 t2+20 t+25

31. 5 x2−125 32. −2 x2+18

33. a3−a b2 34. x2 y− y

35. 12 a2+36 a+27 36. −5 y2+50 y−125

37. x3−2 x2 y+x y2 38. x3 y+2 x2 y2+x y3

Unit 4 Lesson 4 – Factoring a x2+b+c with a=1

Task 4.4

Name______________________________

Date_________ Hour_______

Tic-Tac-But-No-ToePart 1: In the following tic tac’s there are four numbers. Find the relationship that the two numbers onthe right have with the two numbers on the left.

Observations:

1. What did you find?

2. Did it follow the pattern every time?

-90 10

1 -9

36 -6

-12 -6

-36 -6

0 6

-30 -6

-1 5

-49 7

0 -7

120 30

34 4

-81 9

0 -9

24 -6

-10 -4

-72 24

21 -3

16 4

8 4

-6 -3

-1 2

49 -7

-14 -7

Part 2: Use your discoveries from Part 1 to complete the following Tic Tac’s.

Did your discovery work in every case?

Can you give any explanation for this?

9

10

16

-10

6

7

-35

2

4

-5

45

14

6

-5

-3

-2

-15

2

-6

-5

-72

-1

72

-38

-36

5

-22

9

Unit 4 Lesson 4 – Factoring a x2+b+c with a=1

Notes 4.4

Notes

Factor each trinomial completely, if possible.

1. b2+8b+7 2. m2+m−90

3. n2−10 n+9 4. m2+2 m−24

5. k 2−13 k+40 6. z2−4 z+24

7. 2 n2+6n−108 8. 5 n2+10 n+20

9. 4 v2−4 v−8 10. 2 p2+2 p−4

11. If the only factors of a polynomial are 1 and itself, then the polynomials is _______________.

Additional Notes/Examples:

Unit 4 Lesson 4 – Factoring a x2+b+c with a=1

Ready, Set, Go! - Assignment 4.4

http://goo.gl/GdDfw

Name______________________________

Date_________ Hour_______

Ready1. State whether each of the following statements is true or false.

a. x2−6 x+9= ( x−3 )2 ________________________

b. x2−8 x−9=( x−8 ) ( x−9 ) ___________________________

c. x2−10 xy+9 y2=( x− y ) ( x−9 y ) __________________________

d. x2+ x+1=( x+1 ) ( x+1 ) ________________________

e. x2+ xy+20 y2=( x+5 y ) ( x−4 y ) _________________________

f. x2+1=( x+1 )2 __________________________

2. How can you check if you have factored a trinomial correctly?

3. What should you always look for first when attempting to factor a polynomial completely?

SetFactor each polynomial completely. If the polynomial is prime, say so.

4. y2+7 y+10 5. a2−6 a+8

6. m2−10 m+16 7. m2−17 m+16

8. m2+6 m−16 9. w2−8−2 w

10. −16+m2−6 m 11. a2−2 a−12

12. x2+3 x+3 13. 3 y+ y2−10

14. m2+12m+20 15. t 2+30t +200

Go!Factor each polynomial completely. If the polynomial is prime, say so.16. 2 k2+22 k+60 17. 4 v2−30 v+40

18. 6 v2+66 v+60 19. 2 p2+2 p−4

20. 4 v2−4 v−8 21. 5 v2−30v+40

22. 5 n2+10n+20 23. 2 n2+6n−108

Factoring Matching Activity

Name______________________________Date_________ Hour_______

Cut out each pair of quadratic equations and match each equation to its equivalent form.

Standard Form Factored Form

1. y=x2+3 x+2 a. y=(x−1)(x+3)

2. y=x2+2 x−3 b. y=(x−3)(x+1)

3. y=x2−1 x−6 c. y=(x+1)(x−5)

4. y=x2+x−6 d. y=(x−3)(x+2)

5. y=x2+5 x+4 e. y=(x−2)(x−5)

6. y=x2+2 x−3 f. y=(x+2)(x+4)

7. y=x2−x−6 g. y=(x+1)(x+4)

8. y=x2+6 x+5 h. y=(x+1)(x+2)

9. y=x2−4 x−5 i. y=(x+2)(x−6)

10. y=x2−4 x−12 j. y=(x+1)(x+5)

11. y=x2+6 x+8 k. y=(x+2)(x−3)

12. y=x2−5 x+6 l. y=(x+2)(x+5)

13. y=x2−2 x−3 m. y=(x−2)(x+3)

14. y=x2+7 x+12 n. y=(x+3)(x−1)

15. y=x2+7 x+10 o. y=(x−3)(x−2)

16. y=x2−7 x+10 p. y=(x+3)(x+4)

Factoring Review Assignment Lesson 4.1-4.4

Name______________________________

Date_________ Hour_______

Review Assignment Covering Sections 4.1-4.4

Write an equivalent form of each expression by factoring each polynomial completely.

1. x4−x3 2. 2 w2−162 3. 6 w4−54 w2

4. −a3−100 a 5. x3−2 x2 6. x3+7 x2

7. 4 r2+9 8. t2+4 z2 9. x2 w2+9 x2

10. w2−18 w+81 11.w2+30 w+81 12. 6 w2−12w−18

13. ax+ay+cx+cy 14. y3+ y2−4 y−4 15. −2 x2−10 x−12

16. −a3−2 a2−a 17. 32 x2−2 x4 18. 20 w2+100 w+40

19. w3−3 w2−18w 20. 18 w2+w3+36 w 21. 9 y2+1+6 y

22. 2 a2+1+3 a 23. 3h2 t+6 ht +3 t 24. 6 x3 y+30 x2 y2+36 x y3

25. 3 x3 y2−3 x2 y2+3 x y2 26. 5+8 w+3w2 27. ac+xc+aw2+x w2

28. a3+ab+3b+3 a2 29.−4w3−16 w2+20 w 30.−3 y3+6 y2−3 y

Unit 4 Lesson 5 – Factoring a x2+b+c with a ≠1

Notes 4.5

Notes on factoring trinomials when a ≠ 1.

Rewrite each of the following polynomials in an equivalent form by factoring each completely.

1. 3 x2+7 x+2 2 . 3 p2−2 p−5

3. 3 n2−8 n+4 4. 2 v2+11v+5

5. 2 n2+3 n−9 6. 2 n2+5 n+2

7. 36x2 + 12x + 1 8. 6x2 + 26x + 24

9. 9 k 2+66 k+21

Unit 4 Lesson 5 - Factoring a x2+b+c with a ≠1

Ready, Set, Go! - Assignment 4.5

http://goo.gl/Ma7Y0

Name______________________________

Date_________ Hour_______

Ready1. True or False.

a. 3 x2+4 x−15=(3 x+5)(x−3) ____________________

b. 4 x2+4 x−3=(4 x−1)(x+3) _____________________

c. 4 x2−4 x−3=(2 x+1 ) (2 x−3 ) _____________________

d. 4 x2+8 x+3=(2 x+1 ) (2x+3 ) _____________________

2. Explain trial-and-error factoring.

3. What should you always first look for when factoring a polynomial?

SetFactor each polynomial completely. If prime, say so.

4. 6 w2+5w+1 5. 4 x2+11 x+6

6. 2 x2−5x−3 7. 2 a2+3 a−2

8. 4 x2+16 x+15 9. 6 m2−m−12

10. 12 x2+5 x−2 11. 30 b2−b−3

12. 6 a2+a−5 13. 2 x2+15 x−8

14. 3 a2+20 a+12 15. 4 x2−5 x+1

16. 4 x2+7 x+3 17. 7 u2+11u−6

18. 6 y2−7 y−20 19. 5 m2+13 m−6

Unit 4 Lesson 6 – Additional FactoringNotes 4.6

Factor each polynomial completely, if possible. If a polynomial is prime, say so.

1. x4−9 2. y8−14 y4+49

3. x2 m− y2 4. x10−9

5. y8−4 6. a6+10 a3+25

7. z12−6 z6+9 8. x6−8

9. a2n−1 10. b4 n−9

Unit 4 Lesson 6 – Additional FactoringTask 4.6

Name______________________________

Date_________ Hour_______

1. Which of the following are not perfect square trinomials? Explain.

a¿ 4a6−6a3 b4+9b8 b¿1000 x2+200 ax+a2

c ¿900 y4−60 y2+1 d ¿36−36 z7+9 z14

2. Which of the following is not a difference of two squares? Explain.

a¿16 a8 y4−25 c12 b¿a9−b4

c ¿ t90−1 d ¿ x2−196

3. Factor each polynomial and explain how you decided which method to use.

a¿ x2+10 x+25

b¿ x2−10 x+25

c ¿ x2+26 x+25

d ¿ x2−25

e ¿x2+25

Unit 4 Lesson 6 – Additional FactoringReady, Set, Go! - Assignment 4.6

Name______________________________

Date_________ Hour_______

Factor each completely. If prime, say so.1. y6−27 2. a2r+6 ar+9

3. u6 n−4 u3 n+4 4. x6−2 x3−35

5. x4+7 x2−30 6. a20−20 a10+100

7. b16+22 b8+121 8. x10−100

9. y8−9 10. y6−8

1. 13. 25.

2. 14. 26.

3. 15. 27.

4. 16. 28.

5. 17. 29.

6. 18. 30.

7. 19. 31.

8. 20. 32.

9. 21. 33.

10. 22. 34.

11. 23. 35.

12. 24.

Answer Sheet for Factoring Around the Room

Unit 4 Lesson 7 – Factoring Review Assignment 4.7

Name______________________________

Date_________ Hour_______

Factor completely. If not factorable, write prime.

1. 3 x2+15 x 2. x3 y4−x2 y 3

1._____________________

2._____________________

3. 3 x−4 xy+6 y 4. x3−2 x2+4 x−8 3._____________________

4._____________________

5. 2 x3−8 x2+3 x−12 6. 2 x3+x2+x+1 5._____________________

6._____________________

7. x2+4 x+3 8. x

2−5x−24 7._____________________

8._____________________

9. x2+x−30 10. 2 x2+x−1 9._____________________

10.____________________

11. 3 x2−10 x−8 12. 4 x2−15 x−4 11.____________________

12.____________________

13. x2−9 14. 2 x2−50 13.____________________

14.____________________

15. x2+16 16. x2−144

15.____________________

16.____________________

17. 3 x3+3 18. 8 x3+6417.____________________

18.____________________

19. x2−81 y2 20. (x – 4)3(x – 2)2 – 3(x – 4)2(x – 2)2

19.____________________

20.____________________

Circle your answer.

21. Mrs. Rich is trying to carpet her room. If her room is a square and has an area of , write expressions that represent the length and width of her room.

Unit 4 Lesson 8 – Factoring Simple Quadratic Expressions over the Complex Number System

Task/Notes 4.8

Name________________________________________

Date_________ Hour________

1. Consider the polynomial x2−1. How would you factor this polynomial?

2. Consider the polynomial x2+1.How would you factor this polynomial?

3. Can you solve the polynomialx2+1=0?

4. How could you factor the polynomial x2+1? ()()

Be sure to distribute your factorization below to be sure it is in fact an equivalent form.

To be able to factor x2+1 you need to use imaginary numbers. To factor such examples you are factoring over the set of Complex Numbers.

5. Factor x2+25 over the set of complex numbers.

6. Factor 49 x2+144 y2 over the set of complex numbers.

In the next Unit we will discuss how to write more complicated quadratic expressions in factored form, problems with complex roots. Today’s lesson will focus on more simple examples.

7. Factor x4−1 over the set of complex numbers.

8. How would the factorization be different if you were to factor x4−1 over the set of real numbers? Explain.

9. Factor 9 x2+100 over the set of the complex numbers.

10. Factor 64 y2+121 x2 over the set of complex numbers.

Unit 4 Lesson 8– Factoring Simple Quadratic Expressions over the Complex Number System

Ready, Set, Go! Assignment - 4.8

Name__________________________________

Date_______ Hour________

ReadyFactor each over the complex number system.

1. x2+1 2. x4−1

3. 4 x2+1 4. 16 y2+9

5. 25 m2+16 n2 6. 36 a2+49

7. −100+ y2 8. 121 x2+4

9. 144 y2+169 z2 10. 225 a4−4

Factor each over a) The real number system: b) The complex number system:

Set

The real number system.

11a. x4−1

12a. y4−4

13a. z4−9

14a. a4−16

15a. b4−25

16a. c4−36

Go!17a. x4− y4

18a. 16 a2−25b2

19a. 49−100 a4

20a. x8−1

The complex number system.

11b. x4−1

12b. y4−4

13b. z4−9

14b. a4−16

15b. b4−25

16b. c4−36

17b. x4− y4

18b. 16 a2−25b2

19b. 49−100 a4

20b. x8−1

Factoring Cut-outs – Cut out each puzzle piece and reassemble so that the expressions and their factored forms match up.

x2+4x-21

x2+6x+9 (x)

(3x) (x+4)(x-1)

x2+3x-4

x2

(x+5)

x2-4

x2-64

(x-2)(x+2)

x2+7x+10 (x+10)(x+2)

x2+8x+7

x2+6x (x)(x)

(x-2)(x+4)(x+2)(x+10) x2+9x+20 x2+20x+100 (x+5)(x+4)

x2-7x-18 (x+3) 2

x2+10x+25 (3x)(x) x2-1 (5x)(3x) x2+4x+4 x2+3x-4

(x+10)(x+2) (x+4)(x-1) (x+4)(x+3) (x+7)(x-3)x2+2x-8

x2-5x 3(x+2)

(x+10) 2

x

2+7x+10 3x+6

(x-10)(x-4)

x(x+6)

x2+12x+2

0

x2+6x (x+2) 2

x2-4x-5

(x+5)(x+2) (x+6)

(x+10) x2+9x+20x2-14x+40

15x2 (x+2)(x-9)

x2+12x+20

x2+3x-4

3x2

(x-8)(x+8)

x(x+1)

(x+9)(x-6)

x2+x (5x)(3x)

(x+1)(x-5)

x2+20x+100

3x+6 (x+5) 2

(x+7)(x+1)

Factoring Review Worksheet # 1The G.C.F.

Name_____________________________________

Date___________ Hour_________

Directions: Find the missing factor of each expression below. Write the factor in the blank in the term. Then find your answer in the Answer Bank and write its corresponding letter in the blank before the problem. When you have finished, write the letters in order, starting with the first problem, to complete the statement at the end of the activity.

1. _______ 98a=¿ _________ (7a )

2. _______ 15a=¿_________ (5)

3. _______ 12 a2=¿_________ (6 a)

4. _______ 3 a2 b=¿_________ (a)

5. _______ 18 ab=¿__________ (9a)

6. _______ 27a2b2=¿_________ (3ab)

7. _______ 6 a+6b=¿__________ (a+b)

8. _______ 21 a+28=¿___________ (3a+4)

9. _______ 42a+54b=¿___________ (7a+9 b)

10. _______ 12 a+3a2=¿___________ (4+a)

11. _______ 15a2+12a+30=¿_________ (5a2+4a+10)

12. _______ 2 b2−2b=¿___________ (b−1)

Finding the Missing Factor

13. _______ 12 a2 b+18 a2 b+6 a=¿_________ (2 ab+3 ab+1)

14. _______ 3 a3 b2−3a2 b3+3 ab4=¿_________ (a2−ab+b2)

15. _______ a3b+2a2b2+4ab3=¿__________ (a2+2ab+4 b2)

16. _______ −10ab+4 ab3+14b4=¿_________ (−5a+2a b2+7b3)

17. _______ a2b7−2 a2 b6+a5 b3=¿__________ (b4−2 b3+a3)

18. _______ −7 a+49 a2−14=¿___________ (−a+7 a2−2)

19. _______ 15 a2−27 a=¿__________ (5 a−9)

20. _______ 12 a3+30 a2−12a=¿___________ (2 a2+5a−2)

Answer Bank

A. 3a O. 2b S. 6 L. 6a M. a2b3 F. 14 I. 7

Y. 3 ab2 P. 3 C. 2a T. 3ab R.9ab N. ab

You can check your answers for this activity by multiplying each factor.

If one of the ___ ___ ___ ___ ___ ___ ___ ___ ___ ___

___ ___ ___ ___ ___ ___ ___ ___ ___ ___, you must use the Distributive Property.

Factoring Review Worksheet # 2Difference of Squares

Name_____________________________________

Date___________ Hour_________

Directions: Factor each polynomial, if possible, and write the factors in the space after the polynomial.

1. x2−36=¿ _________________________________________

2. x2−64=¿ _________________________________________

3. x2−1=¿ _________________________________________

4. x2+16=¿ _________________________________________

5. 4 x2−121=¿ _________________________________________

6. 2 x2−25=¿ __________________________________________

7. x4−4=¿ __________________________________________

8. 16 x2−49=¿ __________________________________________

9. x2 y2−9=¿ __________________________________________

10. x3−144=¿ __________________________________________

11. 25 x2 y2−36=¿ _________________________________________

12. 10 x4−81=¿ _________________________________________

13. x6 y 4−169=¿ __________________________________________

14. x6 y 8−100=¿ __________________________________________

15. 8 x4 y2−25=¿ __________________________________________

Factoring Review Worksheet # 3Factoring Trinomials 1

Name_____________________________________Date___________ Hour_________

Directions: Factor each trinomial, if possible, and write your answer in the space provided.1. x2+5 x+6=¿ _____________________________________________________________________

2. x2−6 x−7=¿ _____________________________________________________________________

3. x2−12 x+32=¿____________________________________________________________________

4. x2−4 x+4=¿ _____________________________________________________________________

5. x2−9 x+8=¿ _____________________________________________________________________

6. x2+ x−20=¿_____________________________________________________________________

7. x2−x−30=¿_____________________________________________________________________

8. x2−16 x+60=¿____________________________________________________________________

9. x2−3 x−28=¿_____________________________________________________________________

10. x2−2 x−15=¿ ___________________________________________________________________

11. x2+3 x+2=¿_____________________________________________________________________

12. x2−15 x+36=¿___________________________________________________________________

13. x2−6 x+5=¿_____________________________________________________________________

14. x2−12 x+27=¿___________________________________________________________________

15. x2−6 x−40=¿____________________________________________________________________

16. x2−21 x+108=¿_________________________________________________________________

Factoring Review Worksheet # 4Factoring Binomials and Trinomials

Name_____________________________________Date___________ Hour_________

Directions: Factor each polynomial and write the factors in the space after the polynomial.

1. x2−x−6=¿_____________________________________________________________________

2. x2−9 x+20=¿_____________________________________________________________________

3. x2+9 x+14=¿_____________________________________________________________________

4. x2−9 x+8=¿_____________________________________________________________________

5. 3 x2+16 x+16=¿__________________________________________________________________

6. x2−13 x+40=¿____________________________________________________________________

7. x2−6 x+8=¿_____________________________________________________________________

8. x2+2x−3=¿_____________________________________________________________________

9. x2−x−2=¿_____________________________________________________________________

10. x2+8 x+16=¿ ___________________________________________________________________

11. 2 x3−16 x2=¿_____________________________________________________________________

12. 4 x+28=¿_____________________________________________________________________

13. 2 x2−x−6=¿_____________________________________________________________________

14. 2 x3+2 x2=¿_____________________________________________________________________

15. x2−11 x+24=¿___________________________________________________________________

16. 6 x2+42 x=¿_____________________________________________________________________

Directions: Factor each trinomial. Hint: one factor of each polynomial is a factor of the polynomial in the next problem. Always check your work.

17. 3 x2−11 x−4=¿_______________________________________________________________

18. 6 x2−x−1=¿_______________________________________________________________

19. 2 x2+13 x−7=¿_______________________________________________________________

20. x2−x−56=¿ _______________________________________________________________

21. x2−5 x−24=¿ _______________________________________________________________

22. 4 x2+11 x−3=¿_______________________________________________________________

23. 2 x2+11 x+15=¿______________________________________________________________

24. 6 x2+11x−10=¿______________________________________________________________

25. 12 x2+x−6=¿ _______________________________________________________________

26. 4 x2+15 x+9=¿_______________________________________________________________

27. x2+12x+27=¿ _______________________________________________________________

28. 5 x2+52 x+63=¿______________________________________________________________

29. 10 x2−x−21=¿_______________________________________________________________

30. 12 x2−28 x+15=¿_____________________________________________________________

31. 18 x2−3 x−10=¿ ______________________________________________________________

Factoring Review Worksheet # 5Reviewing Factoring Skills

Name_____________________________________

Date___________ Hour_________

Directions: Read each statement and decide whether it is true or false. If it is true, write “true”. If it is false, write “false” and provide an example or an explanation that will make the statement true.

1. 10 is the GCF of 20 and 40. ________________________________________________________

______________________________________________________________________________

2. 36 is a square number. ________________________________________________________

______________________________________________________________________________

3. x ( x2+1 )=x3+x . ________________________________________________________

______________________________________________________________________________

4. x2−16 cannot be factored. _______________________________________________________

______________________________________________________________________________

5. 2 x4 is the GCF of 10 x4+12x2+2. ________________________________________________

______________________________________________________________________________

6. (x+3) is a factor of x2+ x−12. __________________________________________________

______________________________________________________________________________

7. (2 x2+1 ) ( x2−1 )=2x4−x2−1.________________________________________________

______________________________________________________________________________

8. x+1is a factor of x2+1. ________________________________________________________

______________________________________________________________________________

9. The product of two binomials is always a trinomial. ____________________________________

10. 51 is a prime number. ___________________________________________________________

______________________________________________________________________________

11. 2 x+1 is a factor of 6 x2−5 x−4._________________________________________________

______________________________________________________________________________

12. 16 x3 is a perfect square. ________________________________________________________

______________________________________________________________________________

13. 1 may be a GCF. _______________________________________________________________

______________________________________________________________________________

14. x+3 is a factor of x4+3 x3+ x2+3 x . _____________________________________________

______________________________________________________________________________

15. 2 x2−5x−12cannot be factored. _________________________________________________

______________________________________________________________________________

16. 2 x3+4 x2−16 x is factored completely as x ( 2x2+4 x−16 ) .__________________________

______________________________________________________________________________

17. x3+27is the sum of two cubes. ___________________________________________________

______________________________________________________________________________

18. x2+1x+3 cannot be factored. ___________________________________________________

______________________________________________________________________________

19. ( x3−8 )=( x−2 ) ( x2−2 x+4 ). _________________________________________________

______________________________________________________________________________

20. The number of false statements in this exercise is a factor of 20. __________________________

Factoring Review Worksheet # 6Factoring Polynomials Completely

Name_____________________________________

Date___________ Hour_________

Directions: Factor each polynomial completely. See “flow chart” in textbook to help you.

1. 2 x2−6 x+4=¿ _______________________________________________________________

2. x3+3x2−4 x=¿_______________________________________________________________

3. 3 x2−12x−36=¿_______________________________________________________________

4. 16 x2+16 x+4=¿_______________________________________________________________

5. 3 x2−27=¿____________________________________________________________________

6. −x+4 x3=¿_______________________________________________________________

7. 25 x4−100 x2=¿_______________________________________________________________

8. x4−1=¿_______________________________________________________________

9. 15 x2−9 x−6=¿_______________________________________________________________

10. 12 x2+38 x+16=¿______________________________________________________________

11. 30 x3+21 x2+3x=¿_____________________________________________________________

12. 2 x6−8=¿_______________________________________________________________

13. x4− y4=¿_______________________________________________________________

14. x2 y−9 y+3 x2−27=¿__________________________________________________________