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Algebra I

Unit 3: Operations with Monomials & Polynomials

Date Day CW Topicpgs. # Unit 3 Packet

1 1 & 2 Exponents & Their Properties (Operations w/Monomials) 3.1 p. 3

2 4 & 5 Zero and Negative Exponents 3.2 p. 6 & 7

3 8 & 9 Scientific Notation 3.3 p. 10 & 11

4 12 & 13Combining Like Terms (Adding & Subtracting Monomials & Polynomials)

3.4p. 14 & 15

5 16 & 17Quiz 3: 3.1 - 3.3 Multiplying a Polynomial by a Monomial

3.5p. 18 & 19

6 20 & 21Multiplying Polynomials 3.6 p. 22 & 23

7 24 & 25Dividing By Monomials 3.7 p. 26 & 27

8 Pre-Test 3.7 p. 29 & 30

9 Test 3.8 Test

Homework

Name: ____________________________________ Date: __________________

Algebra 1, Unit #6 – Quadratic Algebra – L1 The Arlington Algebra Project, LaGrangeville, NY 12540

Exponents and Their Properties - Multiplying and Dividing Monomials Algebra 1

When we want to express a product of the same number like3 3 3 3⋅ ⋅ ⋅ we can use a shortcut notation exponent

43

base

The exponent tells how many times the base is used as a factor in the product. Exercise #1: Write each of the following in the form of an expanded product.

(a) 3x = (b) 54 = (c) ( )32x = (d) ( )2

5x + =

Exercise #2: Express each of the following with an equivalent expression involving exponents. (a) z z z z z⋅ ⋅ ⋅ ⋅ = (b) 6 6 6⋅ ⋅ = (c) 2 2 2 2a a a a⋅ ⋅ ⋅ = (d) x x y y y⋅ ⋅ ⋅ ⋅ = (e) ( )( )( )x y x y x y+ + + =

Exercise #3: Consider the product shown below:

4 3x x⋅

(a) Write both parts of this product as extended products.

4x = 3x =

(b) Write the product 4 3x x⋅ as an expanded product and in terms of an equivalent expression involving an exponent.

Exercise #4: Express each of the following products as a single variable raised to a power.

(a) 2 7x x⋅ (b) 6 2x x⋅ (c) 2 3 4a a a⋅ ⋅ (d) 2 6 10y y y⋅ ⋅


We also must be able to divide monomial expressions that have the same base. The key to understanding how this process works is the following:

Exercise #5: Consider the quotient 5

3.

x

x

(a) Fill in the following: 5 2 _______.x x= ⋅ (b) Rewrite the numerator of the quotient

using (a).

Exercise #6: Fill in the blanks in the box below: Exercise #7: Write each product or quotient in its simplest form. (a). 4 4a a⋅ =

(b) 2 7x x⋅ =

(c) 4 23 3⋅ =

(d) 2 6y y⋅ = (e) 3 42 2 2⋅ ⋅ =

(f) 5 2y y y⋅ ⋅ =

(g) ( )23x =

(h) 8

2

10

6

y

y=

(i) 3

2

z

z=

(j) 2

7

b

b=

(k) 2 5

8 2

x y

x y=

(l) ( )32

4

x

x=

Any quantity divided by itself, except for zero, is equal to 1.

EXPONENT PROPERTIES

For any real numbers a and b,

1. _________a bx x x⋅ = 2. _________a

b

xx

x=

(c) Rewrite the quotient as the product of two fractions, one of them being equal to 1.

(d) Simplify the quotient using the Multiplicative Identity Property of Real Numbers.

Name: ____________________________________ Date: __________________


Exponents and Their Properties - Multiplying and Dividing Monomials Algebra 1 Homework

Skill Express the product with exponents. 1. a a a b b⋅ ⋅ ⋅ ⋅ = 2. ( )( )( )2 2 2x x x =

3. ( )( )2 2x x y y⋅ =

Express the product in simplest form.

4. 3b b⋅ =

5. 4 9y y⋅ =

6. 2 3 4x x x⋅ ⋅ =

7. 4n n⋅ = 8. y y⋅ =

9. 4 2a a⋅ =

10. 3 7x x⋅ = 11. 4 4z z⋅ = Express the quotient in simplest form.

13. 5

4

x

x=

14. 10

4

a

a=

15. 5

8

x

x=

16. 6

12

y

y=

17. 13 5

2 9

x y

x y=

18. 5 3

8 10

8

4

x y

x y=

19. 4

4

y

y=

Reasoning Simplify.

20. c

d

xc d

x= >

21. ( ) ( )32 2z z z =

22. 4 2a ax x⋅ =

23. 3 6 4

5 2

x x x

x x

⋅ ⋅ =⋅

24. 2 3a a

a

y y

y

⋅ =

25. ( )

3 4

22

x x

x

⋅ =

26. 4 5x y⋅ = Determine True or False for each. State the reason for your answer.

27. 4

22

1x

x=

28. 5

23

42

2=

Name: ____________________________________ Date: __________________


Exponents and Their Properties – Multiplying and Dividing Monomials Additional Practice Problems

Skill Find the product. 1. 5 3x x⋅ = 2. 2y y⋅ = 3. 4 2 6y y y⋅ ⋅ = 4. a a a a⋅ ⋅ ⋅ =

5. 6 4a a⋅ =

6. 3 7t t⋅ = 7. 5 3r r⋅ =

8. 4 5x x⋅ = 9. 3 8y y⋅ = 10. t t⋅ =

11. 2 2w w w w⋅ ⋅ ⋅ = 12. 5 3p p p⋅ ⋅ = 13. 6 3x x x⋅ ⋅ = 14. 8 3h h⋅ = 15. 4 2x x⋅ = 16. 2y y⋅ = 17. x x x⋅ ⋅ =

18. 23 17m m⋅ =

19. 15 5b b⋅ =

Find each quotient.

20. 34

4=

21. 3x

x=

22. 5

2

x

x=

23. 4

2

y

y=

24. 2

2

t

t=

25. 5

3

z

z=

26. 9

5

z

z=

27. 9

3

t

t=

28. 35

5=

29. 3

2

p

p=

30. 3

3

6

6=

31. 12

10

x

x=

Reasoning Simplify.

32. 34

4

x

x=

33. 3 2t t

t

⋅ =

34. 3

2

a b

b

• =

35. ( )2

2

ab

ab=

36. 4 2

3 5

x y

x y=

37. 4 2

3

4

2

x y

xy=

38. a ax x⋅ =

39. a

b

x

x=

40. 4 7

2 3

x y z

x yz=

41. 4 2

4 2

x y

x y=

42. 5

2 2

xy

x y=

43. 2

3 2

x y

x y=

Name: ____________________________________ Date: __________________


Zero and Negative Exponents Algebra 1

In our last lesson we learned how to simplify products and quotients of monomials using laws of exponents with positive integers. But, zero and negative exponents are also possible.

Exercise #1: Recall that a

a bb

xx

x−= .

(a) Using this exponent law, simplify each of the following.

4

4

x

x=

10

10

x

x=

7

7

y

y=

(b) What must each of these quantities equal, assuming none of the variables equals zero? Exercise #2: Simplify each of the following:

(a) 0125 = (b) ( )02y = (c) 05x = (d) ( )302x =

We can investigate negative exponents in a very similar fashion to the zero exponent. The key is to define a negative exponent in such a way that our fundamental rules for exponents don’t need to change.

Exercise #3: Consider the quotient 2

5

x

x.

Exercise #4: Rewrite each expression in simplest terms without the use of negative exponents.

(a) 24− = (b) 2x− = (c) 32− = (d) 10y− =

(a) Write this quotient using the exponent law from Exercise #1.

(b) Write this quotient in its simplest form without a negative exponent.


Exercise #5: Rewrite each of the following monomials without the use of negative exponents.

(a) 2

1

x− = (b) 5

1

y− = (c) 3

1

x− (d) 5

7

y

x

−

− =

Exercise #6: Which of the following is equivalent to 2 5

5 3

x y

x y

−

− − ?

(1) 3

8

x

y (3) 3 8x y

(2) 8

3

y

x (4)

3 8

1

x y

Exercise #7: Rewrite the following expressions without negative or zero exponents.

(a) 24− = (b) 04 = (c) 24−− = (d) 21− = (e) 21−− = (f) ( )01− =

(g) 03x = (h) 32x− = (i) 5

6

x− = (j) 72x− = (k) 3

2

a

d

−

− = (l) 3 2

4

r t

s

−

− =

Exercise #8: Evaluate each of the following expressions using the values 1a = − , 2b = and 3c = . Use the STORE feature on your calculator to aid you.

(a) ( )2 cab

−− = (b) ( ) 1abc

− = (c)

02

2 15

a

b c

− =

(d) b c aa b c− − − =

NEGATIVE AND ZERO EXPONENTS

If a is any integer and 0x ≠ then

(1) 1aa

xx

− = (2) 1 a

ax

x− = (3) 0 1x =

Name: ____________________________________ Date: __________________


Zero and Negative Exponents

Algebra 1 HomeworkSkills For problems 1 through 36, rewrite without zero or negative exponents. 1. 34− =

2. 25−− =

3. 05 =

4. 210− =

5. 34−− =

6. 42− =

7. 2

1

2− =

8. 0

1

4=

9. ( ) 23

−− =

10. 03x =

11. 45x− =

12. 5

3

x

y− =

13. 4

3

a

b

−

− =

14. 0 22x y−− = 15. 32− =

16. ( )02 516x y− =

17. 03− =

18. 0 38x y− =

19. ( ) 33

−− =

20. 1

1

2

− =

21. 2

1

2

− =

22. 1

1

3

− =

23. 61− =

24. ( )05− =

25. ( ) 21

−− =

26. 12−− =

27. ( ) 12

−− =

28. ( ) 22

−− =

29. ( ) 122−−− =

30. 3 2

4 1

2

4

x y

x y

−

− − =

31. 3 4a b− =

32. 2

4

a

b

−=

33. 2

32

x

y− =

34. 3

4

3x

y−− =

35. 0 3

2

x y

z

−=

36. 1 42x y− − = Use the STORE feature on your calculator to help evaluate the following. 37. 3y− for 2y =

38. 3 1 for

2y y− =

39. 4 1 12 for 2,

3x y x y− − = =

40. ( ) 23 for 4x x

−+ = −

41. for 2, 2yx x y− = − =

42. ( )04 2 4 2

3 7 for ,x y x y −= =

43. 2 4

5 3 for ,y yx x x y− −= =


Reasoning Fill in the missing for each of the following.

44. 1

39

= □

45. 2 14− =□

46. 21

25−=□

47. 122

−=□

48. 2 16− =□

49. 1

1010,000

=□

50. 1

381

= □

51. 1

464

= □

Write the answer to each of the following as a single number.

52. ( )0 3[ 1 5 2 ]− + + =

53. ( )2

113 1

2− + − =

54. 3

1 83

3

−− + =

55. Evaluate each of the

following products:

(a) 3 32 2−⋅ =

(b) 2 25 5−⋅ =

(c) 4 410 10− ⋅ =

(d) a ax x−⋅ = 56. Which of the following

is correct?

(a) 33

12

2x

x− =

(b) 33

22x

x− =

Explain why the other choice is incorrect.

True or False

57. 1

12

2

− =

58. 1

4 4

3 3

− = −

59. ( ) 2 12

4−− =

60. 3 2 2

3 2 3 5

2 2x y y

a x a x

−− −=

61. 7 43 3 27−⋅ =

62. ( ) 326

1a

a

−− =

63. ( )04 0− =

64. 3 3 02 2 2 2− ⋅ ⋅ =

65. 2 1 5

3 2 3

x y x

x y y

−

− =

Find the value of x that makes each statement true. 66. 4 122 2 2x ⋅ =

67. 2 95 5 5x− ⋅ =

68. ( )2 104 4x =

69. 23 27x− =

70. ( )2 2 44 3 5 1x−⋅ ⋅ =

71. 2 6 12

4x+ =

Name: ____________________________________ Date: __________________


Combining Like Terms Algebra 1

We have already seen the process of combining like terms when solving linear equations. In this lesson we will broaden our understanding of what constitutes like terms and how to combine them. First, we review the reasoning process behind combining like linear terms. Exercise #1: Fill in the blanks for each with the real number property that justifies the particular step. (1) 6 2 3 4 6 3 2 4x y x y x x y y+ + + = + + + (1)

(2) ( ) ( )6 3 2 4

9 6

x y

x y

= + + += +

(2)

Exercise #2: Combine each of the following like terms using the Distributive Property.

(a) 2 7x x+ = (b) 2 25 2x y x y− + = (c) 2 25 6 3 4x x x x+ + + = Clearly like terms are those monomials in an expression that have the same variables raised to the same power. We should be able to combine them mentally by first identifying like terms and then summing all coefficients of those terms. Exercise #3: Combine all like terms in the following expressions.

(a) 8 4 6 7x x+ − − (b) 2 24 20 3 5y y− + + (c) ( )3 2 3 5w w− −

(d) 2 22 3 7 5 8 3x x x x+ − + − + (e) 2 23 9 6 4 2 8x x x x− + − + − − Exercise #4: Which of the following expressions cannot be simplified? (1) 3 6x x+ (3) 3 6x y+

(2) 6 3y y− (4) 2 22 7x x+


We will oftentimes be asked to combine terms either in sums or differences. Differences can be particularly tricky because subtraction is not commutative, meaning the order in which you do the subtraction will change the result.

Exercise #5: Which of the following represents the sum of ( ) ( )2 23 3 8 and 5 4 2x x x x− + − + + ?

(1) 28 10x x− − + (3) 22 10x x− +

(2) 22 10x x− + + (4) 28 7 6x x− +

Exercise #6: From ( )27 8 3x x+ − subtract( )24 5 6 .x x− +

(1) 23 3 3x x+ + (3) 211 13 3x x+ +

(2) 23 3 9x x− + + (4) 23 13 9x x+ −

Exercise #7: When ( )24 8 3x x− − is subtracted from ( )2 2 1x x− + the result is

(1) 23 6 4x x− + + (3) 25 6 4x x+ +

(2) 23 6 2x x+ − (4) 23 6 2x x− − −

Additional Classroom Practice Simplify by combining. 1. 5 2 8 7x y x y+ − + = 2. ( ) ( )2 4 5 9x x− + + =

3. ( ) ( )2 5x y x y− − + =

4. 2 23 4 2 2 5 1x x x x+ + + − − =

5. 2 23 5 1 4 2 3x x x x− + − + + =

6. ( )2 25 8 3 2x x x x+ − − =

7. ( ) ( )2 22 3 7x x x x− − − =

8. From 24 2 3x x+ − subtract 2 3.x −

9. Add 2 2

2 2

4 2 3

3

a ab b

a ab b

+ +

− − +

10. Subtract 2

2

3 4 5

5 2 1

y y

y y

+ −

− +

11. ( )( )2 24 6 3 2 4x x x x− − − + =

12. How much less than 25 3 2x x− + is 22 5x + ?

Name: ____________________________________ Date: __________________


Combining Like Terms Algebra 1 Homework

Skill Combine as indicated. 1. 9 7 2 4x x+ + − = 2. ( )4 3 3 2x x− − − =

3. ( ) ( )2 4 3 5x x x− + − =

4. ( )2 24 3 5 2 3 5t t t t− + + − + − =

5. ( ) ( )12 9 5 3t t− − − =

6. ( ) ( )6 4 8a b a b− − + =

7. ( )2 2 2 23 3 4 2xy xy x y x y xy+ − − + =

8. ( ) ( )2 23 4 4x x x x− + − − =

9. 2 4 2 43 5 6y y y y+ − + =

10. 2 2 2 25 4 6 3 2x xy y x xy y xy− + − + + + =

11. 2 2 23 5 2 4 1 5 2a a a a a a+ + + − − − + =

12. 2 2 2 2 2 2 2 22 6 5r s t v r s t v− + − − = 13. 8 5 8 5a c b c a b+ + − − − =

14. ( )4 10 5 2y y y− + − + − =

15. combine: 2 4, 3 2, 5x x x+ − + +

16. combine: 2 25 25; 6 9a a a a− + − + +

17. combine: 3 2 2 2 3 24 5 , 2 4y y z yz yz y y z− + − − +

18. Sum:2 2 2 2 26 3 , 4 4 , 6 3 5x y y xy xy x y− + − − − +

19. Subtract 5 3x y+ from 4x y+ 20. From 5 3x y+ subtract 4x y+ Applications 21. Represent the perimeter of a square whose

side length is given by the binomial 4 6x + . 22. Represent the perimeter of a rectangle

whose width is y and whose length is the binomial 2 7y − .

23. Write the length of the arc �ABC as a binomial involving x, y, and z.

A

B

C

2xy-z

3xy+2z


24. Express the perimeter of the triangle as a binomial.

25. The perimeter of a triangle is given by the

expression 212 4 15x x− + . Find the third side of the triangle if the other two sides

measure 24 3x + and 5 4x − . Reasoning 26. Recall that two expressions are additive

inverses if their sum is equal to zero. Find the additive inverse for each of the following:

(a) 7 4x − (b) 2 4 5c c− + 27. What is 3 5a b− + decreased by 9 2a b+ 28. By how much does 4 3x − exceed

7 5x + ?

29. How much greater than 2x xy− is

25 10x xy+ ? 30. What expression must be added to

23 5 4x x− + to give the result

27 5 6x x− − ? 31. From the sum of 6 5x − and 2 4x + subtract

3 9x −

32. Subtract the sum of 22 3 4x x− + and 2 2 1x x+ − from 26 2 1x x− +

33. ( ) ( ) ( )2 24 6 9 2 4y y y y− + + − − =

34. ( ) ( )2 2 2 24 2 3 4 4 2x y xy xy x y xy x y+ − − − +

35. ( ) ( )3 2 3 23 5 6 3 8 8x x x x x+ − + − + +

x-4y 5x+8y

2x+3y

Name: ____________________________________ Date: ________________


Multiplying a Polynomial by a Monomial Algebra 1

In the previous lessons, you’ve worked with monomials and their exponent properties. In this lesson we will begin to work with polynomials, or expressions that contain more than one monomial. The most common polynomials are binomials (those with two monomial terms) and trinomials (those with three monomial terms). First, we review the important real number properties associated with multiplying monomials. Exercise #1: Fill in the blanks for each of the following with the real number property that justifies the particular step. Exercise #2: Simplify each of the following products using real number properties like in Exercise #1.

(a) ( )( )32 3x xy = (b) ( )( )3 54 2y z yz− = (c) ( )( )2 23 2abc a b =

Clearly, we would like to be able to do this multiplication without going through each of these steps. It should be clear from the last exercise that you can simply multiply the coefficients together and then add powers on like bases. Exercise #3: Find the following products.

(a) ( )( )2 3 54 2x y xy = (b) ( )( )2 25 2r s rs = (c) ( )( )3 2 23 6pt p t− − =

We now need to be able to multiply polynomials by monomials. You have actually done this before, as the following exercise will illustrate. Exercise #4: Rewrite the following without parentheses by applying the Distributive Property. (a) ( )5 4x + = (b) ( )3 2 7x− − = (c) ( )6 3x− − =

( ) ( )

( ) ( ) ( )

2 3 2 2 3 2

2 3 2

5 3

3 5 3 5

3 5

15

x y x y x x y y

x x y y

x y

= ⋅ ⋅ ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ ⋅ ⋅

=

(1) (1)

(2) (2)

(3) (3) Exponent Property of Multiplication


Multiplying monomials with variables over polynomials uses the Distributive Property in the same way. Exercise #5: Rewrite the following products without parentheses by applying the Distributive Property.

(a) ( )2 3 4x x + (b) ( )23 2 5 4x x+ − (c) ( )2 23 2ab a b ab+

Additional Classroom Exercises Find the product:

1. ( )( )2 22x y x y− =

2. ( ) ( )33 2yt y t =

3. ( ) ( )2 3 35 4ab c a b c =

4. ( )( )4 3 22x y xy− − =

5. ( )( )3 2 4 27 3p r t pr t =

6. ( )( )3 2 44 3r x r x− =

7. ( )3 2 5x − =

8. ( )6 4x− + =

9. ( )3 2 9x x + =

10. ( )5 3x x − =

11. ( )2 4 7x x− − + =

12. ( )2 5 7x x + =

13. ( )25 4 2 4ab a b ab a+ − =

14. ( )2 24 5 4 2x x x− + =

15. ( )3 2 22 3 4xy x xy y+ − =

Distribute and Combine Like Terms: 16. ( ) ( )5 3 2 1 3x x x x− − − =

17. ( ) ( )3 2 7 2 2 7x x x− + − =

18. ( ) ( )5 2 5x x x+ − + =

19. ( ) ( )3 3 3x x x− − − =

20. ( ) ( )x x y y x y+ − + =

Name: ____________________________________ Date: ________________


Multiplying a Polynomial by a Monomial Algebra 1 Homework

Skill Find the product: 1. ( ) ( )6 2xy z− =

2. ( )( )25 5a a− =

3. ( )( )37x x− =

4. ( )( )2 3 32 4r s r s− =

5. ( )( )214 24d ad− =

6. ( )22x− =

7. ( )223a b =

8. ( )5 2 8x x − =

9. ( )1

36 3x x − =

10. 3 1

4 28 x − − =

11. ( )22 17 214 p p− =

12. ( )3 22 6 3ab a b− =

13. ( )22 2 4x x− + =

14. ( )23 2 8x x− − =

15. ( )3 2 9x x − =

16. ( )25 2 3 7x x x− + =

Distribute and Combine Like Terms: 17. ( )4 3 2 5x− + =

18. ( ) ( )2 23 7 6 2x x x x− + − − − =

19. ( ) ( )2 23 8 5 2 4 10x x x x+ − − − + − =

20. ( ) ( )7 4 7x x x+ + + =

21. ( ) ( )2 3 4 5 3 4x x x− + − =

22. ( ) ( )a a b b a b− + − =

23. ( ) ( )22 4 3 5 1x x x x+ − + =

24. ( ) ( )4 4 3 3 4 3x x x+ + + =


Applications 25. If the length of a square can be represented by the monomial 3x then:

26. The width of a rectangle is represented by w. The length is three more than twice the width.

Reasoning 27. Simplify each of the following if possible. If not possible, explain why.

(a) 3 4x x+ = (b) 3 4x x⋅ = (c) 4 3x x− = (d) 4

3

x

x=

28. Determine each of the following products by writing them out in an expanded product form. The

first is done as an illustration for you.

(a) ( )

( )

32 2 2 2

2 2 2

3 2

6

x x x x

x

x

x

+ +

= ⋅ ⋅

=

=

=

(b) ( )33x = (c) ( )25x = (d) ( )34x =

29. Fill in the blank for the following Exponent Property:

( ) ( )_________bax x=

(a) Express the perimeter of the square as a monomial in terms of x.

(b) Express the area of the square as a monomial in terms of x.

(a) Express the perimeter of the rectangle as a binomial in terms of w.

(b) Express the area of the rectangle as a binomial in terms of w.

Name: ____________________________________ Date: __________________


Multiplying Polynomials Algebra 1

In the last lesson we worked extensively with multiplying polynomials by monomials. In this lesson we will generalize this process so that we may multiply polynomials by polynomials. The first exercise will illustrate the real number properties associated with this process. Exercise #1: Fill in the blanks below with the real number property that justifies each step.

Exercise #2: Using real number properties, find the products given below. (a) ( ) ( )2 4 3 1x x+ − = (b) ( )( )7 5x x+ − = (c) ( )( )2 3 4 6y y− −

Multiplying two linear binomials is such an important skill that a mnemonic has been developed to help remember it: FOIL – Multiply the First, Outer, Inner, and Last terms of the two binomials together and then combine the like terms. Exercise #3: Multiply the following binomials together either using a method as in Exercise #2 or by “FOILing” the two binomials. (a) ( )( )4 1x x+ + = (b) ( )( )3 5y y+ − = (c) ( )( )2 7 3 2x x− + = (d) ( )( )5 5x x− + =

Exercise #4: Which of the following is equivalent to ( )24x − ?

(1) 2 16x + (3) 2 8 16x x− −

(2) 2 16x − (4) 2 8 16x x− +

( )( ) ( ) ( )

( )2

2 4 4 2 4

4 2 2 4

4 2 2 4

4 2 2 4

6 8

x x x x x

x x x x

x x x x

x x x

x x

+ + = + + +

= ⋅ + ⋅ + ⋅ + ⋅

= ⋅ + ⋅ + ⋅ + ⋅

= ⋅ + + + ⋅

= + +

(1)

(3)

(2)

(4)


We can also multiply polynomials together that have more than just two terms. Each term in the first polynomial must multiply each term in the second polynomial for the distribution property to occur. Exercise #5: Find the following product by distributing the binomial over the trinomial.

( )( )22 3 3 4 9x x x− − + =

Since multiplication of these higher powered polynomials can become confusing, it is helpful to use a multiplication table to carry out the product. Exercise #6: Use the following table to help evaluate the following product.

( ) ( )22 3 4 7x x x− − + =

Additional Classroom Exercises 1. ( )( )5 2x x+ − =

2. ( ) ( )2 5 3x x+ + =

3. ( )( )2 3x x− + =

4. ( ) ( )3 5 2 4x x− + =

5. ( )25x − =

6. ( )22 3x − =

7. ( )23 1x + =

8. ( )( )6 6x x+ − =

9. ( )( )2 1 2 1x x+ − =

10. ( ) ( )4 5 4 5x x+ − =

11. ( ) ( )23 2 2 5 1x x x− + −

x

2−

23x 4x− 7

Name: ____________________________________ Date: __________________


Multiplying Polynomials Algebra 1 Homework

Skill Find each of the following products in simplest form. 1. ( )( )2 3x x− − =

2. ( )( )6 1y y+ − =

3. ( )( )5 3a a+ + =

4. ( )( )4 5r r+ + =

5. ( )( )2 3 5x x− + =

6. ( )( )2 9 3x x− + =

7. ( )( )7 2y y− − =

8. ( ) ( )2 5 3 7a a+ + =

9. ( ) ( )3 8x x− + =

10. ( ) ( )5 9x x+ − =

11. ( )( )3 10 5x x+ − =

12. ( )( )5 7x x+ + =

13. ( )( )6 4x x− + =

14. ( )( )3 2 4 3x x− + =

15. ( )( )9 8x x+ + =

16. ( )( )2 3 5x x− − =

17. ( )( )4 4x x+ − =

18. ( ) ( )7 7x x− + =

19. ( ) ( )2 2y y+ − =

20. ( ) ( )4 3 4 3x x+ − =

21. ( ) ( )5 5x x+ − =

22. ( ) ( )3 3y y− + =

23. ( )( )4 1 4 1x x+ − =

24. ( )22x + =

25. ( )26x − =

26. ( )24 1x + =

27. ( )23 2x − =

28. ( )26 1x + =

Applications 29. If the side length of a square is given by the

binomial ( )4 3x − then which of the following

gives the square’s area?

(1) 8 6x + (3) 216 24 9x x+ +

(2) 216 9x − (4) 216 24 9x x− +


Reasoning 30. Find the products of the following polynomials:

(a) ( )( )23 5 2 4 3x x x+ − + = (b) ( )( )23 2 9x x x− + + =

31. Consider the following expression: ( )32x +

(a) Rewrite the expression as a product of three binomials. (b) Evaluate this product by multiplying the last two binomials in part (a) to form a trinomial and then

multiply this trinomial by the first binomial. 32. Rewrite the following expression without the use of parentheses. Keep in mind that you must

multiply the binomials together first and then perform the subtraction.

( )( ) ( )( )3 5 1 4x x x x− − − + − =

Name: ____________________________________ Date: __________________

Algebra 1, Unit #9 – Measurement – L6 The Arlington Algebra Project, LaGrangeville, NY 12540

Scientific Notation Algebra 1

Exercise #1: Use your calculator to evaluate each of the following expressions and record the output: (a) 42.3 10× = (b) 62.3 10× = (c) 32.3 10−× = d) Explain any patterns you notice. e) Which do you expect 42.3 10−× to be equal to? (i) 2300 (ii) 23000 (iii) .0023 (iv).00023 f) What is the output when 42.3 10−× is entered in your calculator?

Scientific Notation is used to express numbers that are very large or very small and is written: 10xa × , where 1 10a≤ < and x is some integer. Exercise #2: Express each in Scientific Notation and write the equivalent expression in aEx form.

The same number can be expressed in various ways: 42.3 10 2.3 -4 .00023E−× = =

(a) 92960000 (Miles between Earth and Sun) (b) 300 million (Population of the US) (c) 0.000000000753 (kg-mass of a dust particle)

Note: You can check all of your answers by placing your calculator in Sci MODE. All outputs in this mode (to any problem) are given in aEx form.


Exercise #3: Express each of the following without scientific notation. (a)2.57 8E (b) 95.28 10−× (c) 79.3 10×

Exercise #4: Jean wants to evaluate the quotient 5

5

2.3 10

2.3 10

××

.

(a) What is the answer to this division problem? (b) Jean types this expression into her calculator and gets the result shown below. Explain why she did

not get the correct answer. (c) Evaluate 2.5 5 2.5 5E E÷ using your calculator. Do you get the correct answer now?

Exercise #5: Perform the indicated operation. Express answers in scientific notation. (a) 6 28 10 2 10× ÷ × (b) 2 75 10 3 10−× ⋅ × (c) The distance from Earth to the Sun is 79.296 10× miles. The distance from Mars to the sun is

81.413 10× miles. How many miles further from the Sun is Mars than Earth? Exercise #6: The Population density of a region is the number of people per square mile and is calculated by finding the ratio of the population to the land area. The Earth’s population is 96.5 10× and has a land area of 75.8 10× square miles. Find the population density. Include units.

When performing operations with numbers expressed in scientific notation, use the form aEx in your calculator or use parentheses to ensure proper order of operations!

The form aEx is entered into the calculator as: a→ 2nd →EE→ x

Name: ____________________________________ Date: __________________


Scientific Notation Algebra 1 Homework

Skills 1. Express in Scientific Notation. (a) 20720 (b) 4 million (c) 0.00008 2. Write each number without scientific notation. (a) 78.42 10−× (b) 3.9216 7E (c) 151 10−× 3. Perform the indicated operation; write your answer in scientific notation.

(a) 5

3

1.76 10

3.2 10−

××

(b) 6 22.5 10 1.5 10−× ⋅ × (c) 9 75.5 10 8.47 10× + ×

Applications 4. For this problem, reference Exercise #6 from the lesson. Find the population density for each

below. Include units. (a) NYC: Population 8.1×106 (b) Lagrange, NY: Population 1.5×104

Land Area 3.03×102 mi2 Land Area 39.7 mi2 5. A human red blood cell is approximately 9×10−3mm in diameter. Approximate, in mm, the width

of 48.2 10× cells if they are positioned side by side in a line.


6. The area of the United States is approximately 65 times larger than the state of New York. If NY is 5.4475 × 104 square miles, approximate the number of square miles contained in the entire US?

7. The diameter of a U.S. quarter is 22.41 10−× m and the diameter of Earth is 71.2753 10× m. How

many quarters would it take, placed side by side, to reach across the Earth’s diameter? Reasoning 8. Julie has left her calculator in school and still has a few homework problems to do. She was

assigned the following problems:

(a) 5 34.2 10 2 10× ⋅ × (b) 5

3

4.2 10

2 10

××

Her mom gave her the following answers and then explained her method.

(a) 8.4×108 (b) 2.1×102

Explain the method that Julie’s mom used by completing the following rules:

10 10x ya b× ⋅ × =

10

10

x

y

a

b

× =×

Name: ____________________________________ Date: __________________

Algebra 1, Unit #7 – Rational Algebra – L2 The Arlington Algebra Project, LaGrangeville, NY 12540

Writing Equivalent Rational Expressions Algebra 1

We know from middle school mathematics that two different fractions can have the same value when reduced. These are called equivalent fractions. The keys to writing and recognizing equivalent fractions are the following two properties of real numbers:

Exercise #1: Write three fractions that are equivalent to 3

2 by multiplying by one in various forms.

Exercise #2: Consider the rational expression 2

3

x +.

(c) Verify that these expressions are equivalent by entering the answer that you wrote in (a) into 1Y

and your answer to (b) in 2Y into your calculator. Fill in the table for selected values of x.

Exercise #3: Which of the following is not equivalent to 6

3

x

x

−+

?

(1) 2 12

2 6

x

x

−+

(3) 3 6

3 3

x

x

−+

(2) 2

2

6

3

x x

x x

−+

(4) 6

3

x

x

− +− −

1 as long as 0 and 1a a b ba

= ≠ ⋅ =

(a) Write an equivalent rational expression by

multiplying by 2

2.

(b) Write an equivalent rational expression by

multiplying by x

x.

x Y1 Y2

1

4

7

10


Simplifying Rational Expressions with Monomial Denominators – Often rational expressions contain monomial denominators (only one term in the denominator). Simplifying these types of rational expressions is an important skill.

Exercise #4: Consider the rational expression 5 2

3 5

2

6

x y

x y.

Exercise #5: Using a process similar to what was used in Exercise #4, simplify each of the following rational expressions by writing it as the product of two fractions, one of which is equivalent to one.

(a) 6

2

4

12

x

x (b)

3

8

10

25

x

x (c)

5 2

3 7

8

12

x y

x y

We would, of course, like to do this simplifying without writing out these two fractions. You should be able to use the exponent law for division to simplify these more quickly. Exercise #6: Simplify each of the following.

(a) 3 6

2 4 3

a b c

a b c (b)

3 2

4

12

18

x y

xy (c)

3 10

9 2

6

2

x y

x y

Because division, like multiplication, distributes over addition and subtraction we can also simplify rational expressions that have polynomials in the numerator. Exercise #7: Simplify each of the following.

(a) 3 26 12

3

c c

c

− (b)

3 210 25 30

5

x x x

x

+ + (c)

2 5 6 3

2 2

4 2

2

x y x y

x y

−

(a) Write this expression as the product of two fractions, one of which is equal to one.

(b) Write this rational expression in simplest terms.

Name: ____________________________________ Date: __________________


Writing Equivalent Rational Expressions Algebra 1 Homework

Skills

1. Which of the following is not equivalent to the fraction 5

3?

(1) 10

6 (3)

20

12

(2) 5

3

−−

(4) 7

5

2. Which of the following is equivalent to 2

1

x

x

+−

?

(1) 3 2

3 1

x

x

+−

(3) 4 8

4 4

x

x

+−

(2) 10

5

x

x

+−

(4) 2

2

2

1

x

x

+−

3. Written in simplest form the fraction 3 2

5

6

12

x y

xy

− is equal to

(1) 2

32

x

y

− (3) 2 32x y−

(2) 3

2

2y

x (4)

2

3

2x

y−

4. Simplify each of the following rational expressions involving only monomials.

(a) 10

4

a

a (b)

3

9

28

4

x

x

− (c)

12

3

18

6

x

x (d)

412

3

x

x

(e) 2 5

4 10

4

6

a b

a b (f)

3 5

7 2

27

9

x y

x y− (g)

2 5

7 3 8

10

25

a b c

a b c (h)

6 9

2 3

10

4

a b

a b


5. Which of the following is equivalent to 6 4

2

30 15

5

x x

x

−?

(1) 3 26 3x x− (3) 4 26 3x x− (2) 4 24 3x x− (4) 34 3x x− 6. Simplify each of the following rational expressions that contain a polynomial numerator and a

monomial denominator.

(a) 32 20

4

x − (b)

3 28 4

2

x x

x

− (c)

xy x

x

−

(d) 2 28 12

4

x y− (e)

3 4 232 40

8

x z xz

xz

−−

(f) 3 230 24 18

6

n n n

n

− +

(g) 3 220 15 25

5

p p p

p

− + (h)

6 5 4

2

18 9 15

3

x x x

x

+ − (i)

3 5 2 4 56 8 2

2

x y x y xy

xy

− +

Reasoning

7. Consider the rational expression 24 8

2

x x

x

+.

(a) Write the expression in simplest form. (b) Enter both the original expression and your answer from part (a) into

Y1 and Y2 on your calculator and fill in the table. (c) Why are the outputs to the two rational expressions different at 0x = ?

x Y1 Y2

-2

-1

0

1

2

algebra i - pbworks 3 packet.pdf · algebra i unit 3: operations ... quiz 3: 3.1 - 3.3 multiplying...

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