# 6.1E: Exercises - Mathematics

## Practice Makes Perfect

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. (81b^5−24b^3+1)
2. (5c^3+11c^2−c−8)
3. (frac{14}{15}y+frac{1}{7})
4. (5)
5. (4y+17)
1. trinomial
2. polynomial
3. binomial
4. monomial
5. binomial

Exercise 2

1. (x^2−y^2)
2. (−13c^4)
3. (x^2+5x−7)
4. (x^{2}y^2−2xy+8)
5. (19)

Exercise 3

1. (8−3x)
2. (z^2−5z−6)
3. (y^3−8y^2+2y−16)
4. (81b^5−24b^3+1)
5. (−18)
1. binomial
2. trinomial
3. polynomial
4. trinomial
5. monomial

Exercise 4

1. (11y^2)
2. (−73)
3. (6x^2−3xy+4x−2y+y^2)
4. (4y+17)
5. (5c^3+11c^2−c−8)

Determine the Degree of Polynomials

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

Exercise 5

1. (6a^2+12a+14)
2. (18xy^{2}z)
3. (5x+2)
4. (y^3−8y^2+2y−16)
5. (−24)
1. 2
2. 4
3. 1
4. 3
5. 0

Exercise 6

1. (9y^3−10y^2+2y−6)
2. (−12p^4)
3. (a^2+9a+18)
4. (20x^{2}y^2−10a^{2}b^2+30)
5. (17)

Exercise 7

1. (14−29x)
2. (z^2−5z−6)
3. (y^3−8y^2+2y−16)
4. (23ab^2−14)
5. (−3)
1. 1
2. 2
3. 3
4. 3
5. 0

Exercise 8

1. (62y^2)
2. (15)
3. (6x^2−3xy+4x−2y+y^2)
4. (10−9x)
5. (m^4+4m^3+6m^2+4m+1)

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Exercise 9

(7x^2+5x^2)

(12x^2)

Exercise 10

(4y^3+6y^3)

Exercise 11

(−12w+18w)

(6w)

Exercise 12

(−3m+9m)

Exercise 13

(4a−9a)

(−5a)

Exercise 14

(−y−5y)

Exercise 15

(28x−(−12x))

(40x)

Exercise 16

(13z−(−4z))

Exercise 17

(−5b−17b)

(−22b)

Exercise 18

(−10x−35x)

Exercise 19

(12a+5b−22a)

(−10a+5b)

Exercise 20

(14x−3y−13x)

Exercise 21

(2a^2+b^2−6a^2)

(−4a^2+b^2)

Exercise 22

(5u^2+4v^2−6u^2)

Exercise 23

(xy^2−5x−5y^2)

(xy^2−5x−5y^2)

Exercise 24

(pq^2−4p−3q^2)

Exercise 25

(a^{2}b−4a−5ab^2)

(a^{2}b−4a−5ab^2)

Exercise 26

(x^{2}y−3x+7xy^2)

Exercise 27

(12a+8b)

(12a+8b)

Exercise 28

(19y+5z)

Exercise 29

(−4a−3b)

Exercise 30

Exercise 31

Subtract (5x^6) from (−12x^6)

(−17x^6)

Exercise 32

Subtract (2p^4) from (−7p^4)

​​​​​​Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Exercise 33

((5y^2+12y+4)+(6y^2−8y+7))

(11y^2+4y+11)

Exercise 34

((4y^2+10y+3)+(8y^2−6y+5))

Exercise 35

((x^2+6x+8)+(−4x^2+11x−9))

(−3x^2+17x−1)

Exercise 36

((y^2+9y+4)+(−2y^2−5y−1))

Exercise 37

((8x^2−5x+2)+(3x^2+3))

(11x^2−5x+5)

Exercise 38

((7x^2−9x+2)+(6x^2−4))

Exercise 39

((5a^2+8)+(a^2−4a−9))

(6a^2−4a−1)

Exercise 40

((p^2−6p−18)+(2p^2+11))

Exercise 41

((4m^2−6m−3)−(2m^2+m−7))

(2m^2−7m+4)

Exercise 42

((3b^2−4b+1)−(5b^2−b−2))

Exercise 43

((a^2+8a+5)−(a^2−3a+2))

(11a+3)

Exercise 44

((b^2−7b+5)−(b^2−2b+9))

Exercise 45

((12s^2−15s)−(s−9))

(12s^2−16s+9)

Exercise 46

((10r^2−20r)−(r−8))

Exercise 47

Subtract ((9x^2+2)) from ((12x^2−x+6))

(3x^2−x+4)

Exercise 48

Subtract ((5y^2−y+12)) from ((10y^2−8y−20))

Exercise 49

Subtract ((7w^2−4w+2)) from ((8w^2−w+6))

(w^2+3w+4)

Exercise 50

Subtract ((5x^2−x+12)) from ((9x^2−6x−20))

Exercise 51

Find the sum of ((2p^3−8)) and ((p^2+9p+18))

(2p^3+p^2+9p+10)

Exercise 52

Find the sum of
((q^2+4q+13)) and ((7q^3−3))

Exercise 53

Find the sum of ((8a^3−8a)) and ((a^2+6a+12))

(8a^3+a^2−2a+12)

Exercise 54

Find the sum of
((b^2+5b+13)) and ((4b^3−6))

Exercise 55

Find the difference of

((w^2+w−42)) and
((w^2−10w+24)).

(11w−66)

Exercise 56

Find the difference of
((z^2−3z−18)) and
((z^2+5z−20))

Exercise 57

Find the difference of
((c^2+4c−33)) and
((c^2−8c+12))

(12c−45)

Exercise 58

Find the difference of
((t^2−5t−15)) and
((t^2+4t−17))

Exercise 59

((7x^2−2xy+6y^2)+(3x^2−5xy))

(10x^2−7xy+6y^2)

Exercise 60

((−5x^2−4xy−3y^2)+(2x^2−7xy))

Exercise 61

((7m^2+mn−8n^2)+(3m^2+2mn))

(10m^2+3mn−8n^2)

Exercise 62

((2r^2−3rs−2s^2)+(5r^2−3rs))

Exercise 63

((a^2−b^2)−(a^2+3ab−4b^2))

(−3ab+3b^2)

Exercise 64

((m^2+2n^2)−(m^2−8mn−n^2))

Exercise 65

((u^2−v^2)−(u^2−4uv−3v^2))

(4uv+2v^2)

Exercise 66

((j^2−k^2)−(j^2−8jk−5k^2))

Exercise 67

((p^3−3p^{2}q)+(2pq^2+4q^3) −(3p^{2}q+pq^2))

(p^3−6p^{2}q+pq^2+4q^3)

Exercise 68

((a^3−2a^{2}b)+(ab^2+b^3)−(3a^{2}b+4ab^2))

Exercise 69

((x^3−x^{2}y)−(4xy^2−y^3)+(3x^{2}y−xy^2))

(x^3+2x^{2}y−5xy^2+y^3)

Exercise 70

((x^3−2x^{2}y)−(xy^2−3y^3)−(x^{2}y−4xy^2))

Evaluate a Polynomial for a Given Value

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

Exercise 71

Evaluate (8y^2−3y+2) when:

1. (y=5)
2. (y=−2)
3. (y=0)
1. (187)
2. (46)
3. (2)

Exercise 72

Evaluate (5y^2−y−7) when:

1. (y=−4)
2. (y=1)
3. (y=0)

Exercise 73

Evaluate (4−36x) when:

1. (x=3)
2. (x=0)
3. (x=−1)
1. (−104)
2. (4)
3. (40)

Exercise 74

Evaluate (16−36x^2) 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 (−16t^2+75) gives the height of the brush (t) seconds after it was dropped. Find the height after (t=2) seconds.

(11)

Exercise 76

A girl drops a ball off a cliff into the ocean. The polynomial (−16t^2+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 (−4p^2+420p). Find the revenue received when (p=60) dollars.

($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 (−4p^2+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 (−frac{1}{150}x^2+frac{1}{3}x), 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 (0.06x^2+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 (−16t^2+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 (frac{9}{5}c+32) where (c) represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when (c=65°).

(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.

Exercise 86

Ariana thinks the sum (6y^2+5y^4) is (11y^6)

Exercise 87

Jonathan thinks that (frac{1}{3}) and (frac{1}{x}) are both monomials. What is wrong with his reasoning?