Algebra 1

Unit 6B – Factoring

Monday Tuesday Wednesday Thursday Friday

9 A Day 10 B Day 11 A Day 12 B Day 13 A Day

Test – Exponents and Polynomials

Factor GCF and Trinomials

− box method

Factoring Trinomials

Feb. 16 17 B Day 18 A Day 19 B Day 20 A Day

No School

Staff Development

Factoring

Trinomials

Factoring with Patterns

− GCF

− difference of squares

− perfect square trinomials

Retest – CBA #4

Dividing Polynomials

Quiz – Factoring

23 B Day 24 A Day 25 B Day 26 A Day 27 B Day

Divide Polynomials

Quiz – Factoring

Elaboration day

Test – CBA #6

(grade will need to go on the NEXT six

weeks marking period)

1

WARM-UP #_______

Simplify each expression.

1. (x + 4)(x – 6) 2. (10x2 + 5x – 6) – (8x

2 – 2x + 7)

3. 5x2 (2xy – 3x) 4. (2x + 5)(3x + 6)

5. (x2 + y

2) – (-x

2 + y

2) 6.

4 2

2 6

a b

a b

2

Notes – GCF and Factoring

Prime Number – a whole number, greater than 1, whose only factors are 1 and itself

Composite Number – a whole number, greater than 1, that is not prime

Prime Factorization – a whole number expressed as a product of factors are all prime

numbers (i.e. factor tree)

Greatest Common Factor (GCF) – the greatest common factor of two or more integers is

the greatest number that is a factor of all the integers

EX1: State whether each number is prime or composite. If the number if composite, find

its prime factorization (tree).

a. 28 b. 61 c. 112 d. 150

EX2: Find the GCF between two numbers using the calculator.

a. -45, 15

b. 169, 13

c. -20, 440

d. 96, 12, -8

Greatest common factor for the same variable will be LOWEST exponent of that given

variable.

Factoring – to express a polynomial as the product of a monomial and a polynomial

EX3: Find the GCF for each set of monomials.

a. x2, x

5, x

4 b. 49x, 343x

2

c. 4a7b, 28ab d. 96y, 12x, -8y

3

Notes – GCF and Factoring

EX4: Factor each polynomial.

a. 24w + 72z b. 30ab2 + a

2b – 12ac

3

c. x4 – 18x

2 + 22x d. a + 10a

2b

3

e. 88x4 – 11x

7 + 66x

5 f. 14c

3 – 42c

5 – 49c

4

g. 48w2x + 18wx

2 – 36wx h. -x

5 – 4x

4 + 23x

3 – x

6

i. 8x – 7y + w j. 18y2 – 50 k. x

3 + 2x

2 + x

4

Reverse Distribution

Find a monomial and a trinomial whose product is equal to each problem below. Cut and paste it in the

correct place.

Problems Monomials

(GCF) Trinomials

1. 12x2+ 3x – 6

2. 12x4 – 6x

2 + 3x

3. 24x5+ 12x

4 – 4x

3

4. 4x4 – 12x

3 + 6x

2

5. 6x3 – 24x

2 – 12x

6. 10a4b

2 – 5a

3b + a

2b

7. 5a6b

5+ 10a

5b

4 – 15a

4b

3

8. 20a5b

5 + 10a

4b

4 – 30a

3b

3

9. 50a6b

2 – 30a

5b

3+ 10a

4b

4

10. 50a7b

6 – 15a

5b

2 + 25a

3

5

Monomials (GCF) Trinomials

6x (10a2b – 5a + 1)

5a4b

3 (10a

4b

6 – 3a

2b

2+ 5)

10a4b

2 (a

2b

2 + 2ab – 3)

5a3 (4x

2 + x – 2)

3 (x2 – 4x – 2)

3x (2x2 – 6x + 3)

a2b (2a

2b

2 + ab – 3)

10a3b

3 (6x

2 + 3x – 1)

4x3 (5a

2 – 3ab + b 2 )

2x2 (4x

3 – 2x + 1)

6

GCF and Factoring

Factor out the GCF.

1. x3 + x

2 + x 2. 15a + 12b + 6c

3. 8x2 – 18y

2 4. x

2y – 2y

5. z3

+ 4z 6. 4x2 – 4x

7. 15x2 – 50x – 10 8. 12a – 11b

9. 64c3 – 56c

2 + 88c 10. 24x

6y

3 – 32x

3y

2 – 20x

2y

2

11. –2x4 + 24x

2 12. 12x

3y

4 – 40xy

5

Name Date

7

Simplify each expression.

13. (2x + 5xy + 7y) + (3x + 7xy + y) 14. (3x + 2y) – (5x + 6y)

15. (a + b)0 16. 95

71

ba20

ba40−−

−−

17. (2m-4

n3)(-5mn

-7) 18.

19. Find the volume of a cylinder with a diameter of 4x3y and a height of 7x

2y

4.

20. Find the volume of a cube with sides 2b3r

2.

Solve.

21. 4x + 2 = 2(5x – 11) 22. 9 – 4x < 10

2

1b

a−

−

8

WARM-UP #_____

Find the missing information on the given rectangles.

What is the area?

This is the same size rectangle just divided up.

What is the area of the first rectangle?

What is the area of the second rectangle?

What is the area of the whole rectangle?

Write the area of each rectangle inside each box for both of the rectangle below and

answer the questions.

What is the total area of the rectangle?

What is the total area of the rectangle?

What do you notice about all of the rectangles above?

What is special about the length and width?

What is the length and width of all the rectangles?

12

17

12

12 5

3

12 5

9

6

15 2

6

9

EXPLORE

Given the rectangles below, determine the length and width of each rectangle and the area. Rectangles

are not drawn to scale.

Total Area ______ Total Area ______

Length _______ Length _______

Width ________ Width ________

Total Area ______ Total Area ______

Length _______ Length _______

Width ________ Width ________

Total Area ______ Total Area ______

Length _______ Length _______

Width ________ Width ________

Total Area ______ Total Area ______

Length _______ Length _______

Width ________ Width ________

Total Area ______

Length __(x – 1)_____

Width __(2x + 3)______

5 2

5

6

14 12

21

20 12

21

2x 5x

25

10

3x2 2x

20

30x

1 2x

4x

-1

20

2

x

30x

x2

-12

-4

x

10

Notes – Factoring Trinomials

EX1. Recall the box method to multiply two binomials. Multiply (x – 3)(x + 2).

Factors:

Product:

EX2. Find the missing dimension of each trinomial’s box. Fill in the blank cells in each box.

a. a2 + 7a + 10 = (a + 5)( ) b. c

2 – 10c + 21 = (c – 3)( )

c. y2 – 2y – 15 = (y + 3) ( ) d. n

2 + 3n – 28 = (n – 4) ( )

e. How do the quantities you filled in the 2 blank cells relate to the original trinomial?

a

5

a2

10

c -3

c2

21

n -4

n2

-28

y

3

y2

-15

11

Notes – Factoring Trinomials

EX3. Write the numbers that give a sum of –5x and a product of –50x2.

• Standard form:

EX4. Factor each trinomial.

a. x2 + 7x + 10 =

Sums to be: (Middle term: b)

Yield a product of:

(this comes from multiplying the a and c)

b. x2 + 3x – 4 =

Sums to be: (Middle term: b)

Yield a product of:

(this comes from multiplying the a and c)

c. x2 – 64 =

Sums to be: (Middle term: b)

Yield a product of:

(this comes from multiplying the a and c)

12

Notes – Factoring Trinomials

d. 3x2 + 14x + 8 =

Sums to be: (Middle term: b)

Yield a product of:

(this comes from multiplying the a and c)

e. 2y2 – 7y + 6 =

Sums to be: (Middle term: b)

Yield a product of:

(this comes from multiplying the a and c)

f. 6x2 – 21x – 12 =

Sums to be: (Middle term: b)

Yield a product of:

(this comes from multiplying the a and c)

13

Notes – Factoring Trinomials

Factoring Using Algebra Tiles

EX5. Determine the factors of each polynomial.

a. b.

c. d.

14

___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___

11 15 10 11 1 8 16 7 17 12 5 11 2 16 17 6 4 3

___ ___ ___ ___ ___ ___ ___ ___ ___ ___

14 8 7 9 6 4 3 14 16 13

A B C D E

(x – 2)(5x – 8) (2x + 3)(3x – 2) (x – 13)(x + 3) (x + 2)(x – 3) (x – 2)(x + 3)

F G H I J

(x – 8)(2x + 5) (x + 2)(7x + 3) (x – 2)(x + 1) (x – 5)(x + 3) (x + 5)(x – 3)

K L M N O

(x – 7)(x + 7) (x – 2)(x – 8) (x + 2)(x + 8) (x – 7)(x –2) (x + 7)2

P Q R S T

(x – 7)(x + 2) (x + 2)(x – 8) (x – 3)(x + 3) (x – 6 )(2x – 1) (x + 6)2

U V W X Y

(x – 5)(x + 5) (x – 5)(x – 5) (x – 3)2 (x – 3)(x + 20) (x – 6)(x – 2)

Z

(x + 9)(x + 7)

Factoring Trinomials

Directions: Match that answer to the correct letter of the alphabet. Enter that letter of the alphabet on

the blank corresponding to the problem number.

Factor each polynomial – make sure to show your work.

1. x2 – 49 2. x

2 + 12x + 36 3. 7x

2 + 17x + 6

4. x2 – 9x + 14 5. 5x

2 – 18x + 16 6. x

2 – 2x – 15

Name Date

15

7. x2 – 25 8. x

2 – 8x + 12 9. 2x

2 – 13x + 6

10. x2 + x – 6 11. x

2 – 10x – 39 12. 2x

2 – 11x – 40

13. x2 + 17x – 60 14. 6x

2 + 5x – 6 15. x

2 – x – 2

16. x2 + 14x + 49 17. x

2 – 9

Identify the simplified area of each rectangle. Then determine the factors.

18. 19. 20.

16

WARM-UP #______

1. Factor: x2 + 13x + 12

Factors:

What did you notice?

2. Factor: 4x2 – 9

Factors:

What did you notice?

3. Factor: x2 + 6x + 9

Factors:

What did you notice?

17

18

Explain – Factoring with Patterns

Difference of Squares

a2 – b

2 = (a)

2 – (b)

2 = (a + b)(a – b) Conjugate pairs

difference opposite signs

*Warning: a2 + b

2 does not factor

To recognize perfect squares, look for coefficients that are squares of integers and

variables raised to even powers.

EX1: Factor, if possible, using the difference of squares.

a. 4x2 – 9y

2

b. a2 – 16b

2

c. 9x4 – 25y

4

d. u2v

2 – w

2z

2

e. 25m2 + 36n

2

19

Explain – Factoring with Patterns

Perfect Square Trinomials

• a2 + 2ab + b

2 = (a + b)(a + b) = (a + b)

2

• a2 – 2ab + b

2 = (a – b)(a – b) = (a – b)

2

EX2: Factor each of the following.

a. x2 + 6x + 9

b. x2 – 10x + 25

c. a2 + 8a + 16

d. 9a2 – 24a + 16

20

Factoring Patterns

Determine whether each statement is TRUE. If not, find the correct product.

1. (3x + 1)2 = 9x

2 + 6x + 1 2. (m – 4)

2 = m

2 – 16m + 16

3. (5t – 2)2 = 25t

2 – 20t + 4 4. (2n + 7)

2 = 4n

2 + 28n + 49

5. (2b + 3)2 = 4b

2 + 12b + 6 6. (2a + b)

2 = 4a

2 + 4ab + b

2

Factor each polynomial. If it cannot be factored, write prime.

7. t2 – 12t + 36 8. a

2 + 2ab + b

2

9. 4t2 + 20t + 25 10. n

2 – 1

11. 144 – 25n2 12. 16a

2 – 24ab + 9b

2

Name Date

21

13. t2 – 18t + 81 14. 4n

2 – 9

15. 25 + 10t + t2 16. n

2 – 49

17. 49m2 – 16n

2 18. a

2 – 8a + 64

19. 49a2 + 14a + 1 20. 81 – 121n

2

21. Which is the correct factorization of –45x2 + 20y

2?

A. –5(3x + 2y)2 B. 5(3x – 2y)

2

C. –5(3x + 2y)(3x – 2y) D. 5(3x + 2y)(3x – 2y)

22. Challenge Determine the value(s) of k for which each expression is a perfect square

trinomial.

a. 49x2 – 84k + k b. 4x

2 + kx + 9

22

Explore – Dividing Polynomials

Remember when we MULTIPLIED: (using a box)

(x + 2)(x + 6) or (2y + 1)(3y – 4)

So can you now DIVIDE these polynomials: (using a box)

2x

12x8x2

+

++

43y

4y56y2

−

−−

So…..

2x

12x8x2

+

++

43y

4y56y2

−

−−

the quotient is: the quotient is:

x

+2

–4 3y

23

24

Explain – Dividing Polynomials

Dividing is the opposite operation of . Therefore, we will use

the to assist in dividing trinomials when given a trinomial divided

by a binomial.

EX1. Simplify each expression.

a. 1x

5x4x2

−

−+ Quotient:

b. 4x

12x112x2

+

++ Quotient:

c. 13x

5x13x62

−

−+ Quotient:

x

–1

x2

–x

5x

–5

2x2

3x

8x

12

25

Explain – Dividing Polynomials

EX2. Simplify each expression – ON OUR OWN.

a. 7x

14x33x52

+

−+ Quotient:

b. 8x

24x11x2

−

+− Quotient:

c. 12x

9x17x22

−

−+ Quotient:

26

Gingerbread Man

Divide each polynomial. Each answer determines the next location of the traveling gingerbread man.

Determine the path the gingerbread man makes through the school.

1. 9x

27x12x2

+

++ 2.

8x

40x13x2

−

+− 3.

4x

44x7x2

+

−−

4. 6x

42xx2

−

−+ 5.

1x2

4x9x22

+

++ 6.

7x

7x15x22

−

+−

7. 3x2

15x7x22

−

−+ 8.

5x2

5x17x62

+

++ 9.

7x

14x19x32

+

−+

10. 4x3

12x7x122

+

−+ 11.

1x2

3x2x82

−

−+ 12.

4x3

4x15x92

−

+−

Name Date

27

Principal Secretary

Attendance Counselor

Nurse

Cafeteria

Theater

Playground

2

Library

Home Ec

Lab

Computer

Lab

Art Room

5th

Grade

4th

Grade

3rd

Grade

Tea

cher

Wo

rkro

om

K -

2

Tro

phy c

ase

Kindergarte

n

1st Grade

2nd

Grade

Playground

1

Tea

cher

Wo

rkro

om

3 –

5

Front Door

Where’s my

class? •

(x + 3)

•

(x – 5)

•

(x – 11)

•

(x + 7)

•

(x + 4)

•

(x – 7)

•

(x + 5)

•

(3x + 1)

•

(3x – 2)

•

(4x – 3)

•

(4x + 3)

•

(3x – 1)

•

(x + 11)

•

(x – 4)

•

(3x + 2)

•

(2x – 1)

•

(x + 2) •

(x – 2)

•

(x + 1)

•

(x + 6)

28

Review – CBA #6

1. Find the volume of a cube with sides 2b3r

2.

2. Find the volume of a cylinder that has a radius of 5s3t

5 and a height of 2s

2t

4.

3. Find the area of a triangle that has a base of 32mn7 and a height of 3m

4n

3.

4. If a rectangle has an area of 16x7y

4 and a length of 4x

3y, what is its width?

5. Distance (d), rate (r), and time (t) are related by the formula d = rt. If a ball rolls 36p4q

9 feet for

4p2q

3 minutes, what is the rate?

6. Write an expression that best represents the area of a square with sides of 7x4y

3?

7. Find the perimeter and area of the rectangle in terms of n.

3n – 5

2n + 10

Name Date

29

5n – 1 3n + 5

2n

8. Find the perimeter and area of the triangle in terms of x.

Simplify each expression.

9. (-2x + x2) – x(5x – 4) + (9x

2 – 6x) 10. p(2p – 3) + (p – 3)(4p + 1)

11. (3x5)3(2x

7)2 12. (-3x

6)2

13. (3r + 7)2

14.

8 7

2 6 5

12x y z

4x y z

−

− −

15. ( ) ( )

4 2 3 9 0

10 4

2a b 7ab

3a b−

−

−

16. n6 + n + n

6

17. The dimensions of a wall are 7xy feet by 8x2y

3 feet. A picture has

dimensions 2x feet by x2y

4 feet. If the picture is hanging on the wall as

shown, what is the area of the wall not covered by the picture?

18. A pitcher contains 16x5y

4 ounces of water. A mug holds 2x

2y ounces. Leticia pours water from

the full pitcher into mugs. If she filled axby

c mugs, what is the value of

a + b + c?

30

19. Find the area of a circle with radius 6r3s

5 inches.

20. Find the area of a rectangle with side lengths (x2 – 7x) and (2x

2 + 3x + 1).

21. Describe and correct the error in finding the product of the given polynomials.

22. The area of a rectangle is 3x2 – 10x – 8. Find the dimensions (length and width) of the rectangle.

Factor out the greatest common monomial factor.

23. 16a2 – 40b 24. -36s

3 + 18s

2 – 54s 25. 17abc

2 – 6a

2c

31

Completely factor each of the following polynomials.

26. r2 + 2r – 24 27. y

2 – 2y – 15 28. 2x

2 + 12x + 16

29. 2ax2 – 3ax – 35a 30. 4a

2 + 9a – 9 31. 6k

2 + 13k + 6

32. 9x2 – 121 33. k

2 – 49 34. 64u

2 – 25

Identify the simplified area of each rectangle. Then determine the factors.

35. 36. 37.

32