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6.1E: Exercises

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Identify Polynomials, Monomials, Binomials, and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

Exercise 1
  1. 81b524b3+1
  2. 5c3+11c2c8
  3. 1415y+17
  4. 5
  5. 4y+17
Answer
  1. trinomial
  2. polynomial
  3. binomial
  4. monomial
  5. binomial
Exercise 2
  1. x2y2
  2. 13c4
  3. x2+5x7
  4. x2y22xy+8
  5. 19
Exercise 3
  1. 83x
  2. z25z6
  3. y38y2+2y16
  4. 81b524b3+1
  5. 18
Answer
  1. binomial
  2. trinomial
  3. polynomial
  4. trinomial
  5. monomial
Exercise 4
  1. 11y2
  2. 73
  3. 6x23xy+4x2y+y2
  4. 4y+17
  5. 5c3+11c2c8

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

Exercise 5
  1. 6a2+12a+14
  2. 18xy2z
  3. 5x+2
  4. y38y2+2y16
  5. 24
Answer
  1. 2
  2. 4
  3. 1
  4. 3
  5. 0
Exercise 6
  1. 9y310y2+2y6
  2. 12p4
  3. a2+9a+18
  4. 20x2y210a2b2+30
  5. 17
Exercise 7
  1. 1429x
  2. z25z6
  3. y38y2+2y16
  4. 23ab214
  5. 3
Answer
  1. 1
  2. 2
  3. 3
  4. 3
  5. 0
Exercise 8
  1. 62y2
  2. 15
  3. 6x23xy+4x2y+y2
  4. 109x
  5. m4+4m3+6m2+4m+1

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Exercise 9

7x2+5x2

Answer

12x2

Exercise 10

4y3+6y3

Exercise 11

12w+18w

Answer

6w

Exercise 12

3m+9m

Exercise 13

4a9a

Answer

5a

Exercise 14

y5y

Exercise 15

28x(12x)

Answer

40x

Exercise 16

13z(4z)

Exercise 17

5b17b

Answer

22b

Exercise 18

10x35x

Exercise 19

12a+5b22a

Answer

10a+5b

Exercise 20

14x3y13x

Exercise 21

2a2+b26a2

Answer

4a2+b2

Exercise 22

5u2+4v26u2

Exercise 23

xy25x5y2

Answer

xy25x5y2

Exercise 24

pq24p3q2

Exercise 25

a2b4a5ab2

Answer

a2b4a5ab2

Exercise 26

x2y3x+7xy2

Exercise 27

12a+8b

Answer

12a+8b

Exercise 28

19y+5z

Exercise 29

Add: 4a,3b,8a

Answer

4a3b

Exercise 30

Add: 4x,3y,3x

Exercise 31

Subtract 5x6 from 12x6

Answer

17x6

Exercise 32

Subtract 2p4 from 7p4

​​​​​​

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Exercise 33

(5y2+12y+4)+(6y28y+7)

Answer

11y2+4y+11

Exercise 34

(4y2+10y+3)+(8y26y+5)

Exercise 35

(x2+6x+8)+(4x2+11x9)

Answer

3x2+17x1

Exercise 36

(y2+9y+4)+(2y25y1)

Exercise 37

(8x25x+2)+(3x2+3)

Answer

11x25x+5

Exercise 38

(7x29x+2)+(6x24)

Exercise 39

(5a2+8)+(a24a9)

Answer

6a24a1

Exercise 40

(p26p18)+(2p2+11)

Exercise 41

(4m26m3)(2m2+m7)

Answer

2m27m+4

Exercise 42

(3b24b+1)(5b2b2)

Exercise 43

(a2+8a+5)(a23a+2)

Answer

11a+3

Exercise 44

(b27b+5)(b22b+9)

Exercise 45

(12s215s)(s9)

Answer

12s216s+9

Exercise 46

(10r220r)(r8)

Exercise 47

Subtract (9x2+2) from (12x2x+6)

Answer

3x2x+4

Exercise 48

Subtract (5y2y+12) from (10y28y20)

Exercise 49

Subtract (7w24w+2) from (8w2w+6)

Answer

w2+3w+4

Exercise 50

Subtract (5x2x+12) from (9x26x20)

Exercise 51

Find the sum of (2p38) and (p2+9p+18)

Answer

2p3+p2+9p+10

Exercise 52

Find the sum of
(q2+4q+13) and (7q33)

Exercise 53

Find the sum of (8a38a) and (a2+6a+12)

Answer

8a3+a22a+12

Exercise 54

Find the sum of
(b2+5b+13) and (4b36)

Exercise 55

Find the difference of

(w2+w42) and
(w210w+24).

Answer

11w66

Exercise 56

Find the difference of
(z23z18) and
(z2+5z20)

Exercise 57

Find the difference of
(c2+4c33) and
(c28c+12)

Answer

12c45

Exercise 58

Find the difference of
(t25t15) and
(t2+4t17)

Exercise 59

(7x22xy+6y2)+(3x25xy)

Answer

10x27xy+6y2

Exercise 60

(5x24xy3y2)+(2x27xy)

Exercise 61

(7m2+mn8n2)+(3m2+2mn)

Answer

10m2+3mn8n2

Exercise 62

(2r23rs2s2)+(5r23rs)

Exercise 63

(a2b2)(a2+3ab4b2)

Answer

3ab+3b2

Exercise 64

(m2+2n2)(m28mnn2)

Exercise 65

(u2v2)(u24uv3v2)

Answer

4uv+2v2

Exercise 66

(j2k2)(j28jk5k2)

Exercise 67

(p33p2q)+(2pq2+4q3)(3p2q+pq2)

Answer

p36p2q+pq2+4q3

Exercise 68

(a32a2b)+(ab2+b3)(3a2b+4ab2)

Exercise 69

(x3x2y)(4xy2y3)+(3x2yxy2)

Answer

x3+2x2y5xy2+y3

Exercise 70

(x32x2y)(xy23y3)(x2y4xy2)

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Exercise 71

Evaluate 8y23y+2 when:

  1. y=5
  2. y=2
  3. y=0
Answer
  1. 187
  2. 46
  3. 2
Exercise 72

Evaluate 5y2y7 when:

  1. y=4
  2. y=1
  3. y=0
Exercise 73

Evaluate 436x when:

  1. x=3
  2. x=0
  3. x=1
Answer
  1. 104
  2. 4
  3. 40
Exercise 74

Evaluate 1636x2 when:

  1. x=1
  2. x=0
  3. x=2
Exercise 75

A painter drops a brush from a platform 75 feet high. The polynomial 16t2+75 gives the height of the brush t seconds after it was dropped. Find the height after t=2 seconds.

Answer

11

Exercise 76

A girl drops a ball off a cliff into the ocean. The polynomial 16t2+250 gives the height of a ball t seconds after it is dropped from a 250-foot tall cliff. Find the height after t=2 seconds.

Exercise 77

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial 4p2+420p. Find the revenue received when p=60 dollars.

Answer

$10,800

Exercise 78

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial 4p2+420p. Find the revenue received when p=90 dollars.

Everyday Math

Exercise 79

Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of x miles per hour is given by the polynomial 1150x2+13x, where x=30 mph.

Answer

4

Exercise 80

Stopping Distance The number of feet it takes for a car traveling at x miles per hour to stop on dry, level concrete is given by the polynomial 0.06x2+1.1x, where x=40 mph.

Exercise 81

Rental Cost The cost to rent a rug cleaner for d days is given by the polynomial 5.50d+25. Find the cost to rent the cleaner for 6 days.

Answer

$58

Exercise 82

Height of Projectile The height (in feet) of an object projected upward is given by the polynomial 16t2+60t+90 where t represents time in seconds. Find the height after t=2.5 seconds.

Exercise 83

Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial 95c+32 where c represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when c=65°.

Answer

149° F

Writing Exercises

Exercise 84

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

Exercise 85

Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

Answer

Answers will vary.

Exercise 86

Ariana thinks the sum 6y2+5y4 is 11y6

Exercise 87

Jonathan thinks that 13 and 1x are both monomials. What is wrong with his reasoning?

Answer

Answers will vary.

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “identify polynomials, monomials, binomials, and trinomials,” “determine the degree of polynomials,” “add and subtract monomials,” “add and subtract polynomials,” and “evaluate a polynomial for a given value.” The rest of the cells are blank.

b. If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


6.1E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by LibreTexts.

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