Chapter 6 — Exponents and Polynomialscabrillo.edu/~mcaspers/math154/book/ch6.pdf · Math 154 :: Elementary Algebra Chapter 6 — Exponents and Polynomials Section 6.4 — Addition

Chapter 6 — Exponents and Polynomials Caspers

Chapter 6 — Exponents and Polynomials

Section 6.1 —Exponents

Section 6.2 — Negative Exponents

Section 6.3 — Polynomials

Section 6.4 — Addition and Subtraction of Polynomials

Section 6.5 — Multiplication of Polynomials

Section 6.6 — Division of Polynomials

Answers

Math 154 ::

Elementary Algebra

http://cabrillo.edu/~mcaspers/math154/book/answers/ch6.pdf

http://cabrillo.edu/~mcaspers/math154/book/answers/ch6.pdf

Math 154 :: Elementary Algebra Chapter 6 — Exponents and Polynomials

Section 6.1 — Exponents 1 Caspers

212 5

3 10

78

52

x y

x y

29

5

3

2

x

y

22 9

22 5

3

2

x

y

Section 6.1 Exponents

Examples:

Simplify each expression.

a) 2

5

2

5 In this expression, –5 is the base and 2 is the exponent. The parentheses around –5 include the negative

sign in the base.

To simplify:

5 5 25

b) 25

25 In this expression, 5 is the base and 2 is the exponent. There is a multiplication by –1 as well. There are no

parentheses, so the negative sign is not part of the base.

The expression may be written as: 21 5

To simplify: 1 5 5 25

c)

212 5

3 10

78

52

x y

x y

In this expression, it is easiest to simplify the inside of the parentheses before applying the “outside”

exponent.

2

12 5

3 10

3

2

x y

x y

First, simplify the fraction

7852 by dividing 26 into both –78 and 52.

Next, cancel three factors of x from the numerator and denominator, and then cancel five factors of y from

the numerator and denominator. It may help to visualize the cancellation like this:

23

2

x x x x x x x x x x x x y y y y y

x x x y y y y y y y y y y y

Now, apply the outside exponent to each factor in the numerator and denominator, by squaring –3 and 2,

and multiplying the exponents on x and y.

18

10

9

4

x

y

Homework

1. When multiplying like-bases, what operation can you perform on the exponents to simplify the expression?

2. When dividing like-bases, what operation can you perform on the exponents to simplify the expression?

3. How do you determine if the factors in a simplification of a quotient of like-bases are in the numerator or the denominator?

4. When raising a power to a power, what operation can you perform on the exponents to simplify the expression?



5. When simplifying a quotient that is raised to a power, what is often the easiest step to take first?

6. If an expression is raised to the 0 power, what is its value?

7. Simplify each expression.

a) 23

b) 2

3

c) 23

d) 3

3

e) 33


a) 24

b) 3

4

c) 34

d) 2

4

e) 24

Simplify.

9. 4 5x x

10. 6

4

y

y

11. 8

11

z

z

12. 5

4x

13. 018

14. 9

3

7

7

15. 15

15

d

d

16. 8xx

17. 2 54 4

18. 12

z

z

19. 10

42

20. 6y

y

21. 345 2x x

22. 2

345y



23. 2

345

c

c

24. 345

2

d

d

25. 9 3x y x

26. 6

5 4

xy

x y

27. 3

2 8y x

28. 14 5

3 5

w y

w y

29. 10

7

6

4

x

x

30. 2 5 17 3x y x y

31. 0

117x

32. 3

42x

33. 2 5 3x y xy

34. 2

53xy

35. 2 7

13

12

4

c d

c d

36. 14

7

20

40

a

a

37. 3

7

5

20

x

x

38. 109

3

y

y

39. 20 4

20 16

4

16

m n

m n

40. 12 8

5 8

20

5

p q

p q

41.

43

2

2x

y

42. 2

5

3

6

m

n



43.

38

6

16

8

x

x

44.

23

2

20

18

y

y

45.

515 3

15 7

9

18

p q

p q

46.

03

2

14

16

x

x

47.

313

21

9

45

m

m

48.

28 5

9 12

121

44

a c

a c

49. 3 25 4x x

50. 3

4 22 3y y

51. 3

6 210 5m m

52. 4

3 7 53 3x y x y

53. 2 0

8 3 113 14xy x y

54. 2 3

6 103 2n n

55. 3 2

2 44 2y z yz


Section 6.2 — Negative Exponents 5 Caspers

8 6

4 3

12

6

x y

x y

8 4 3

6

2x x y

y

2

5

25

Section 6.2 Negative Exponents

Examples:

Simplify each expression.

a) 2

5

In this expression, –5 is the base and –2 is the exponent. The parentheses around –5 include the negative

sign in the base. The negative exponent moves the factors to the denominator.

To simplify:

1 1

5 5 25

b) 25

In this expression, 5 is the base and –2 is the exponent. There is a multiplication by –1 as well. There are

no parentheses, so the negative sign is not part of the base. The negative exponent moves the factors to the

denominator.

The expression may be written as: 21 5

To simplify: 1 1

15 5 25

c) 8 6

4 3

12

6

x y

x y

In this expression, it is easiest to “move” the factors with negative exponents first. That means that 4x

and 3y move to the numerator, and 6y moves to the denominator. The exponents on these factors

become their opposites during the “move”.

8 4 3

6

12

6

x x y

y

Now, simplify the fraction

126

to –2.

Next, combine the factors of x in the numerator by adding the exponents and cancel three factors of y from the

numerator and denominator.

12

3

2x

y

Homework

1. In your own words, describe what happens to the exponent of a factor when the factor is moved from the numerator to the

denominator of a fraction?

2. In your own words, describe what happens to the exponent of a factor when the factor is moved from the numerator to the

denominator of a fraction?




a) 25

b) 3

5

c) 35

d) 2

5

e) 25


a) 3

4

b) 24

c) 2

4

d) 24

e) 34

Simplify. Final answers should not contain negative exponents.

5. 4x

6. 52

7. 3

1

x

8. 32

9. 3

x

y

10. 2

5

m

n

11. 8 7x x

12. 5

3

y

y

13. 8

11

z

z

14. 4

3x

15. 3

8

6

6

16. 12

12

d

d

17. 16xx

18. 1 74 4

19. 10

z

z

20. 10

45



21. 8y

y

22. 14 3x y x

23. 3

7 9

xy

x y

24. 3

2 5y x

25. 14 5

6 5

w y

w y

26. 10

6

14

4

x

x

27. 2 6 18 3x y x y

28. 0

1117x

29. 3

42x

30. 2 5 3x y xy

31. 2

35xy

32. 3 16

12

24

6

c d

c d

33. 18

9

60

30

a

a

34. 3

9

4

20

x

x

35. 116

3

y

y

36. 22 4

22 12

4

12

m n

m n

37. 10 8

5 8

20

5

p q

p q

38.

45

3

2x

y

39. 2

7

2

6

m

n

40.

312

6

8

4

x

x



41.

23

2

25

15

y

y

42.

514 3

14 6

14

28

p q

p q

43.

03

2

14

6

x

x

44.

311

22

7

35

m

m

45.

26 5

9 12

121

66

a c

a c

46. 3 23 5x x

47. 3

4 22 5y y

48. 3

6 510 2m m

49. 4

4 6 53 3x y x y

50. 2 0

9 7 143 24xy x y

51. 2 3

3 122 3n n


Section 6.3 — Polynomials 9 Caspers

Section 6.3 Polynomials

6.3 — Polynomials Worksheet

Example:

For the polynomial given, find the degree of each term, the degree of the polynomial, the leading term, and the leading coefficient. If

the polynomial has a specific name—monomial, binomial, or trinomial—give that name.

a) 6 3 2 69 12x y x y y

Homework

1. In your own words, define a polynomial.

2. In your own words, define each word: monomial, binomial, trinomial.

3. In your own words, describe how you identify the degree of a polynomial.

4. In your own words, describe how you identify the leading term of a polynomial.

5. In your own words, define each word: constant, linear, quadratic, cubic.

For each polynomial given, find the degree of each term, the degree of the polynomial, the leading term, and the leading coefficient. If

the polynomial has a specific name—monomial, binomial, or trinomial—give that name. You may use a chart like the one below for

each polynomial, but it isn’t necessary, as long as you identify each answer.

6. 4 28x x

7. 3 114 2 17x x x

8. 4 2 2 312 7x y x y y

9. 24 15y y

Arrange each polynomial in descending order. Give the degree of each polynomial and the leading coefficient.

10. 3 2 614 5y y y

11. 10 2 127 12x x x x

Individual Terms

The Degree of

Each Individual

Term

The Coefficient of

Each Individual

Term

The Leading

Coefficient of the

Polynomial

The Degree of

the Polynomial

Specific Name

of the

Polynomial

69x y 7 9

9 7 trinomial 3 2x y 5 –1

61 2y 6 12

Individual

Terms

The Degree of Each

Individual Term

The Coefficient of Each

Individual Term

The Leading Coefficient

of the Polynomial

The Degree of

the Polynomial

Specific Name of

the Polynomial

http://cabrillo.edu/~mcaspers/math154/worksheets/6_3.pdf

http://cabrillo.edu/~mcaspers/math154/worksheets/6_3.pdf


Section 6.4 — Addition and Subtraction of Polynomials 10 Caspers

Section 6.4 Addition and Subtraction of Polynomials

Examples:

Perform the operation. Answers may be written in descending order of power, but it isn’t necessary.

a) 2 27 12 23 5x x x x

To add two polynomials, combine like-terms. Exponents on variables will NOT change. If terms are reordered, take the sign

in front of the term with the term that’s moved.

2 27 12 23 1 5x x x x 2 27 12 5 23x x x x

28 7 23x x

b) Subtract 2 20 4x x

from 211 5x x

When translating a statement that involves “subtract … from…”, polynomials are “switched” between the “math order” and

the “English order”.

This problem becomes: 2 211 5 20 4x x x x

2 211 5 20 4x x x x Distribute the subtraction sign to all of the terms in the parentheses following it.

212 21 9x x Combine like-terms.

Homework

1. In your own words, describe a “like-term”. What must be the same? What may be different?

2. In your own words, describe how to add two polynomials. What changes? What doesn’t change?

Perform the operation. Answers may be written in descending order of power, but it isn’t necessary.

3. 6 3x x

4. 15 3y y

5. 4 7x x

6. 8 11y y

7. 13 5 2 14x x

8. 2 228 4 3x x x

9. 7 45 8 6y xy x y

10. 2 2 213 8 4 6x y xy x y x

11. 3 8 11 2x x

12. 25 60 32m m m

13. 2 22 8 9 7m m m

14. 210 6 14 2a a a

15. 2 215 3y y y

16. 2 23 27 23 14x xy x xy


Section 6.4 — Addition and Subtraction of Polynomials 11 Caspers

17. 225 13 1x x

18. 6 12 8 3x x

19. 224 18 5m m m

20. 2 23 10 11 9n n n

21. 213 7 8 16a a a

22. 2 236 4y y y

23. 2 22 20 28 16x xy x xy

24. 229 54 1x x

25. 2 29 71 1 12 6 4 2 4

m m m

26. 234

18x x x

27. 2 32 4 1 25 5 7 10 3

a a a

28. Add 4x and 25 3 7x x .

29. Add 2 13y y and 4 5y .

30. Subtract 9 4x

from 3 12x .

31. Subtract 22 8y y

from 25 16y y .

32. Subtract 2 2 18m m from 24 11 9m m .

33. Subtract 25 27 19x x

from 2 3 4x x .


Section 6.5 — Multiplication of Polynomials 12 Caspers

Section 6.5 Multiplication of Polynomials

Examples:

Perform each operation. Simplify answers (if not simplified after multiplying).

a) 3 212

4 7y y y

To multiply a monomial by a polynomial, distribute the monomial to each term in the polynomial. Exponents on variables

MAY change.

3 212

4 7y y y 3 2 3 31 1 12 2 2

4 7y y y y y

5 4 3712 2

2y y y

b) 5 3 2x y x y

To multiply two binomials, you may use the “FOIL method”. FOIL stands for the multiplications of the terms: First, Outer,

Inner, and Last. “FOIL-ing” is the same thing as distributing each term in the first polynomial to each term in the second

polynomial.

5 3 2x y x y 5 5 2 3 3 2x x x y y x y y

2 25 10 3 6x xy xy y

2 25 7 6x xy y

c) 2

4x

To square a binomial, multiply it out using the “FOIL method”. There is a pattern. If you recognize it, you are welcome to

use it.

2

4x 4 4x x

4 4 4 4x x x x

2 4 4 16x x x

2 8 16x x

Homework

1. What property is most used when multiplying polynomials?

2. When computing the square of a binomial — for example, an expression of the form 2

a b — what must you remember?

3. Compute each problem.

a) 2

3x

b) 2

3x

c) 2

2 4x

d) 2

24x

e) Using the above problems as examples, in your own words, describe how you can tell when you may use a “shortcut”

exponent rule and when you must “FOIL”?

Perform each operation. Simplify answers (if not simplified after multiplying).

4. 4 2 7x

5. 5 3x

6. 27 6 3y y

7. 10 6x

8. 22 5 12x x

9. 4 8x x

10. 2 3y y



11. 12

18 5x x

12. 2 23 2 6x x x

13. 2 3 22 5 11y y y y

14. 2 24 7xy x xy y

15. 2 25 2mn m mn

16. 5 9y y

17. 3 12d d

18. 3 7x x

19. 312 4

k k

20. 4 5p p

21. 1 6x x

22. 3 1 7y y

23. 2

7x

24. 4 2 9x x

25. 2 21 4 7k k

26. 2 25 4 1y y

27. 2

3x

28. 2 21 8p p

29. 2 212 3 2x x

30. 2 11 3 10k k

31. 3 1xy y

32. 14 10mp mp

33. 7x y x y

34. 2 5a c a c

35. 4 3x y x y

36. 2

2 9x

37. 2 3 2x y x y

38. 2

2 3x y



39. 25 3 10x x x

40. 21 4 7y y y

41. 26 2 8 11x x x

42. 2 22 6 20p p p

43. 2 23 2 1m m m

44. 21 14 2

8 24x x x

45. 2 2x y x xy y

46. 2 2a b a ab b

47. 2 22 3 4x y x xy y

48. Multiply each pair of binomials, and then answer the last question.

a) 2 2x x

b) 5 5y y

c) 1 13 3

p p

d) 2 5 2 5y y

e) 3 1 3 1a a

f) In your own words, describe how the above problems similar before they are multiplied, how they similar after they are

multiplied, and then describe the pattern.


a) 2

1x

b) 2

3y

c) 2

14

k

d) 2

2 7a

e) 2

4 1x




a) 2

2y

b) 2

5x

c) 2

12

a

d) 2

2 1a

e) 2

3 2x




Section 6.6 — Division of Polynomials 15 Caspers

22 5 8

x

x x x

2

2

2 5 8

2

7

x

x x x

x x

x

2

2

7

2 5 8

2

7 8

7 14

22

x

x x x

x x

x

x

2 5 8 2x x x

Section 6.6 Division of Polynomials

Examples:

Perform each operation.

a) 6 4 3

3

8 2 4

4

x x x

x

To divide a polynomial by a monomial, divide the monomial into each term of the polynomial. Notice that after the

division/simplification, there will be the same number of terms in the answer as there were in the polynomial.

6 4 3

3

8 2 4

4

x x x

x

6 4 3

3 3 3

8 2 4

4 4 4

x x x

x x x

3 12 1

2x x

b) 2 5 8 2x x x

To divide a polynomial by a binomial, use polynomial long division. There is another way to divide polynomials by

binomials of degree 1; this method will be covered in the next math course.

The first step is to figure out “what” times the first term, x,

of the divisor 2x will be 2x , the first term of the dividend

2 5 8x x . For this problem that value is x, and it is

written on the top of the long division bar.

Next, multiply that value by the binomial 2x , and write

it below the dividend inside the long division bar, so that

like-terms are lined-up. Subtract that product from the

polynomial 2 5 8x x .

Now, figure out “what” times the first term, x, of 2x

will be 7x , the first term of result of the subtraction above,

7 8x . For this problem that value is 7, and it is the next

term written on the top of the long division bar.

22 is the remainder. In the answer, it is written over the

divisor 2x .

The answer to 2 5 8 2x x x is 22

72

xx

.

You may always check division problems by multiplying the divisor by the quotient and adding the remainder. Doing this

should result in the dividend.

Check: 2 7 22x x 2 5 14 22x x 2 5 8x x

Homework

1. In your own words, describe the “easiest” way to divide a polynomial by a monomial.

2. When you divide a polynomial with n terms by a monomial, how many terms will you have in your quotient (answer)?

3. In your own words, describe how to divide a polynomial by a binomial.




4. 4 12 2p

5. 21 12 7y

6. 5 35

5

n

7. 8 11

4

x

8. 4 60 10k

9. 10 12

6

x

10. 212 4 2x x x

11. 3 220 40 10 5y y y y

12. 3 236 18 12 6p p p p

13. 6 4 2

2

9 45 15

3

x x x

x

14. 3 216 20 3 4k k k k

15. 9 7 8 3

5

6 30 18 15

6

y y y y

y

16. 10 9 5 3 34 14 22 2 2m m m m m

17. 12 10 6 4 23 30 18 12 6x x x x x

18. 6 5 4 3

4

14 21 7 35

7

p p p p

p

19. 2

2

15 3 5

5

y y

y


20. 2 5 13 2y y y

21. 2 8 14

3

k k

k

22. 2 3 12 6x x x

23. 2 10 7 3m m m

24. 2 6 11 1a a a

25. 2 4 21

7

p p

p



26. 23 16 20 4x x x

27. 24 3 20

2

y y

y

28. 26 23 21 2 3k k k

29. 24 8 10 2 1x x x

30. 28 26 15 2 5m m m

31. 25 3 8

5 2

k k

k

32. 3 26 11 10 2p p p p

33. 3 22 10 20 9 4y y y y

34. 2 8 2x x

35. 2 4 2x x

36. 25 7 3x x

37. 3 8 2x x

Documents

Chapter 6 — Exponents and Polynomialscabrillo.edu/~mcaspers/math154/book/ch6.pdf · Math 154 :: Elementary Algebra Chapter 6 — Exponents and Polynomials Section 6.4 — Addition