
6.1E: Exercises


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. $$m^4+4m^3+6m^2+4m+1$$

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

Add: $$4a,\,−3b,\,−8a$$

$$−4a−3b$$

Exercise 30

Add: $$4x,\,3y,\,−3x$$

Exercise 31

Subtract $$5x^6$$ from $$−12x^6$$

$$−17x^6$$

Exercise 32

Subtract $$2p^4$$ from $$−7p^4$$

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.

$$4$$

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.

$$58$$

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?