8.1 Add and Subtract Polynomials

Izabela Mazur

CHAPTER 8 Polynomials

8.1 Add and Subtract Polynomials

Learning Objectives

By the end of this section, you will be able to:

Identify polynomials, monomials, binomials, and trinomials
Determine the degree of polynomials
Add and subtract monomials
Add and subtract polynomials
Evaluate a polynomial for a given value

You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form $a{x}^{m}$ , where $a$ is a constant and $m$ is a whole number, it is called a monomial. Some examples of monomial are $8,-2{x}^{2},4{y}^{3}$ , and $11{z}^{7}$ .

Monomials

A monomial is a term of the form $a{x}^{m}$ , where $a$ is a constant and $m$ is a positive whole number.

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Polynomials

polynomial—A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.

monomial—A polynomial with exactly one term is called a monomial.
binomial—A polynomial with exactly two terms is called a binomial.
trinomial—A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

$\begin{array}{lllll}\text{Polynomial} & b+1 & 4{y}^{2}-7y+2 \qquad & 4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1&\\ \text{Monomial}& 14& 8{y}^{2} & -9{x}^{3}{y}^{5} & -13 \\ \text{Binomial} &a+7& 4b-5 & {y}^{2}-16 & 3{x}^{3}-9{x}^{2} \\ \text{Trinomial} & {x}^{2}-7x+12 \qquad & 9{y}^{2}+2y-8 \qquad & 6{m}^{4}-{m}^{3}+8m & {z}^{4}+3{z}^{2}-1\hfill \end{array}$

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

EXAMPLE 1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

$4{y}^{2}-8y-6$
$-5{a}^{4}{b}^{2}$
$2{x}^{5}-5{x}^{3}-9{x}^{2}+3x+4$
$13-5{m}^{3}$
$q$

Solution

Polynomial

Number of terms

Type

a)

$4{y}^{2}-8y-6$

$3$

Trinomial

b)

$-5{a}^{4}{b}^{2}$

$1$

Monomial

c)

$2{x}^{5}-5{x}^{3}-9{x}^{2}+3x+4$

$5$

Polynomial

d)

$13-5{m}^{3}$

$2$

Binomial

e)

$q$

$1$

Monomial

TRY IT 1.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

a) $5b$ b) $8{y}^{3}-7{y}^{2}-y-3$ c) $-3{x}^{2}-5x+9$ d) $81-4{a}^{2}$ e) $-5{x}^{6}$

Show answer

a) monomial b) polynomial c) trinomial d) binomial e) monomial

TRY IT 1.2

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

a) $27{z}^{3}-8$ b) $12{m}^{3}-5{m}^{2}-2m$ c) $\dfrac{5}{6}$ d) $8{x}^{4}-7{x}^{2}-6x-5$ e) $-{n}^{4}$

Show answer

a) binomial b) trinomial c) monomial d) polynomial e) monomial

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

EXAMPLE 13

Evaluate $5{x}^{2}-8x+4$ when

$x=4$
$x=-2$
$x=0$

Solution

a) $x=4$


Simplify the exponents.
Multiply.
Simplify.

b) $x=-2$


Simplify the exponents.
Multiply.
Simplify.

c) $x=0$


Simplify the exponents.
Multiply.
Simplify.

TRY IT 13.1

Evaluate: $3{x}^{2}+2x-15$ when

$x=3$
$x=-5$
$x=0$

Show answer

a) $18$ b) $50$ c) $-15$

TRY IT 13.2

Evaluate: $5{z}^{2}-z-4$ when

$z=-2$
$z=0$
$z=2$

Show answer

a) $18$ b) $-4$ c) $14$

EXAMPLE 14

The polynomial $-16{t}^{2}+250$ gives the height of a ball $t$ seconds after it is dropped from a 250 foot tall building. Find the height after $t=2$ seconds.

Solution

	$-16{t}^{2}+250$
Substitute $t=2$ .	$-16{\left(2\right)}^{2}+250$
Simplify.	$-16\cdot 4+250$
Simplify.	$-64+250$
Simplify.	$186$
	After 2 seconds the height of the ball is 186 feet.

TRY IT 14.1

The polynomial $-16{t}^{2}+250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall building. Find the height after $t=0$ seconds.

Show answer

$250$

TRY IT 14.2

The polynomial $-16{t}^{2}+250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall building. Find the height after $t=3$ seconds.

Show answer

$106$

EXAMPLE 15

The polynomial $6{x}^{2}+15xy$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with $x=4$ feet and $y=6$ feet.

Solution



Simplify.
Simplify.
Simplify.
	The cost of producing the box is $456.

TRY IT 15.1

The polynomial $6{x}^{2}+15xy$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with $x=6$ feet and $y=4$ feet.

Show answer

$576

TRY IT 15.2

The polynomial $6{x}^{2}+15xy$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with $x=5$ feet and $y=8$ feet.

Show answer

$750

Access these online resources for additional instruction and practice with adding and subtracting polynomials.

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.

1.

a) $81{b}^{5}-24{b}^{3}+1$
b) $5{c}^{3}+11{c}^{2}-c-8$
c) $\dfrac{14}{15}y+\dfrac{1}{7}$
d) 5
e) $4y+17$

2.

a) ${x}^{2}-{y}^{2}$
b) $-13{c}^{4}$
c) ${x}^{2}+5x-7$
d) ${x}^{2}{y}^{2}-2xy+8$
e) 19

3.

a) $8-3x$
b) ${z}^{2}-5z-6$
c) ${y}^{3}-8{y}^{2}+2y-16$
d) $81{b}^{5}-24{b}^{3}+1$
e) $-18$

4.

a) $11{y}^{2}$
b) $-73$
c) $6{x}^{2}-3xy+4x-2y+{y}^{2}$
d) $4y+17$
e) $5{c}^{3}+11{c}^{2}-c-8$

Determine the Degree of Polynomials

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

5.

a) $6{a}^{2}+12a+14$
b) $18x{y}^{2}z$
c) $5x+2$
d) ${y}^{3}-8{y}^{2}+2y-16$
e) $-24$

6.

a) $9{y}^{3}-10{y}^{2}+2y-6$
b) $-12{p}^{4}$
c) ${a}^{2}+9a+18$
d) $20{x}^{2}{y}^{2}-10{a}^{2}{b}^{2}+30$
e) 17

7.

a) $14-29x$
b) ${z}^{2}-5z-6$
c) ${y}^{3}-8{y}^{2}+2y-16$
d) $23a{b}^{2}-14$
e) $-3$

8.

a) $62{y}^{2}$
b) 15
c) $6{x}^{2}-3xy+4x-2y+{y}^{2}$
d) $10-9x$
e) ${m}^{4}+4{m}^{3}+6{m}^{2}+4m+1$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

9. ${7x}^{2}+5{x}^{2}$	10. ${4y}^{3}+6{y}^{3}$
11. $-12w+18w$	12. $-3m+9m$
13. $\text{4a}-9a$	14. $-y-5y$
15. $28x-\left(-12x\right)$	16. $13z-\left(-4z\right)$
17. $-5b-17b$	18. $-10x-35x$
19. $12a+5b-22a$	20. $\text{14x}-3y-13x$
21. $2{a}^{2}+{b}^{2}-6{a}^{2}$	22. $5{u}^{2}+4{v}^{2}-6{u}^{2}$
23. $x{y}^{2}-5x-5{y}^{2}$	24. $p{q}^{2}-4p-3{q}^{2}$
25. ${a}^{2}b-4a-5a{b}^{2}$	26. ${x}^{2}y-3x+7x{y}^{2}$
27. $\text{12a}+8b$	28. $\text{19y}+5z$
29. Add: $4a,-3b,-8a$	30. Add: $4x,3y,-3x$
31. Subtract $5{x}^{6}\text{from}-12{x}^{6}$ .	32. Subtract $2{p}^{4}\text{from}-7{p}^{4}$ .

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

33. $\left(5{y}^{2}+12y+4\right)+\left(6{y}^{2}-8y+7\right)$	34. $\left(4{y}^{2}+10y+3\right)+\left(8{y}^{2}-6y+5\right)$
35. $\left({x}^{2}+6x+8\right)+\left(-4{x}^{2}+11x-9\right)$	36. $\left({y}^{2}+9y+4\right)+\left(-2{y}^{2}-5y-1\right)$
37. $\left(8{x}^{2}-5x+2\right)+\left(3{x}^{2}+3\right)$	38. $\left(7{x}^{2}-9x+2\right)+\left(6{x}^{2}-4\right)$
39. $\left(5{a}^{2}+8\right)+\left({a}^{2}-4a-9\right)$	40. $\left({p}^{2}-6p-18\right)+\left(2{p}^{2}+11\right)$
41. $\left(4{m}^{2}-6m-3\right)-\left(2{m}^{2}+m-7\right)$	42. $\left(3{b}^{2}-4b+1\right)-\left(5{b}^{2}-b-2\right)$
43. $\left({a}^{2}+8a+5\right)-\left({a}^{2}-3a+2\right)$	44. $\left({b}^{2}-7b+5\right)-\left({b}^{2}-2b+9\right)$
45. $\left(12{s}^{2}-15s\right)-\left(s-9\right)$	46. $\left(10{r}^{2}-20r\right)-\left(r-8\right)$
47. Subtract $\left(9{x}^{2}+2\right)$ from $\left(12{x}^{2}-x+6\right)$ .	48. Subtract $\left(5{y}^{2}-y+12\right)$ from $\left(10{y}^{2}-8y-20\right)$ .
49. Subtract $\left(7{w}^{2}-4w+2\right)$ from $\left(8{w}^{2}-w+6\right)$ .	50. Subtract $\left(5{x}^{2}-x+12\right)$ from $\left(9{x}^{2}-6x-20\right)$ .
51. Find the sum of $\left(2{p}^{3}-8\right)$ and $\left({p}^{2}+9p+18\right)$ .	52. Find the sum of $\left({q}^{2}+4q+13\right)$ and $\left(7{q}^{3}-3\right)$ .
53. Find the sum of $\left(8{a}^{3}-8a\right)$ and $\left({a}^{2}+6a+12\right)$ .	54. Find the sum of $\left({b}^{2}+5b+13\right)$ and $\left(4{b}^{3}-6\right)$ .
55. Find the difference of $\left({w}^{2}+w-42\right)$ and $\left({w}^{2}-10w+24\right)$ .	56. Find the difference of $\left({z}^{2}-3z-18\right)$ and $\left({z}^{2}+5z-20\right)$ .
57. Find the difference of $\left({c}^{2}+4c-33\right)$ and $\left({c}^{2}-8c+12\right)$ .	58. Find the difference of $\left({t}^{2}-5t-15\right)$ and $\left({t}^{2}+4t-17\right)$ .
59. $\left(7{x}^{2}-2xy+6{y}^{2}\right)+\left(3{x}^{2}-5xy\right)$	60. $\left(-5{x}^{2}-4xy-3{y}^{2}\right)+\left(2{x}^{2}-7xy\right)$
61. $\left(7{m}^{2}+mn-8{n}^{2}\right)+\left(3{m}^{2}+2mn\right)$	62. $\left(2{r}^{2}-3rs-2{s}^{2}\right)+\left(5{r}^{2}-3rs\right)$
63. $\left({a}^{2}-{b}^{2}\right)-\left({a}^{2}+3ab-4{b}^{2}\right)$	64. $\left({m}^{2}+2{n}^{2}\right)-\left({m}^{2}-8mn-{n}^{2}\right)$
65. $\left({u}^{2}-{v}^{2}\right)-\left({u}^{2}-4uv-3{v}^{2}\right)$	66. $\left({j}^{2}-{k}^{2}\right)-\left({j}^{2}-8jk-5{k}^{2}\right)$
67. $\left({p}^{3}-3{p}^{2}q\right)+\left(2p{q}^{2}+4{q}^{3}\right)$ $-\left(3{p}^{2}q+p{q}^{2}\right)$	68. $\left({a}^{3}-2{a}^{2}b\right)+\left(a{b}^{2}+{b}^{3}\right)$ $-\left(3{a}^{2}b+4a{b}^{2}\right)$
69. $\left({x}^{3}-{x}^{2}y\right)-\left(4x{y}^{2}-{y}^{3}\right)$ $+\left(3{x}^{2}y-x{y}^{2}\right)$	70. $\left({x}^{3}-2{x}^{2}y\right)-\left(x{y}^{2}-3{y}^{3}\right)$ $-\left({x}^{2}y-4x{y}^{2}\right)$

Evaluate a Polynomial for a Given Value

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

71. Evaluate $8{y}^{2}-3y+2$ when: a) $y=5$ b) $y=-2$ c) $y=0$	72. Evaluate $5{y}^{2}-y-7$ when: a) $y=-4$ b) $y=1$ c) $y=0$
73. Evaluate $4-36x$ when: a) $x=3$ b) $x=0$ c) $x=-1$	74. Evaluate $16-36{x}^{2}$ when: a) $x=-1$ b) $x=0$ c) $x=2$
75. A painter drops a brush from a platform 75 feet high. The polynomial $-16{t}^{2}+75$ gives the height of the brush $t$ seconds after it was dropped. Find the height after $t=2$ seconds.	76. A girl drops a ball off a cliff into the ocean. The polynomial $-16{t}^{2}+250$ 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.
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 $-4{p}^{2}+420p$ . Find the revenue received when $p=60$ dollars.	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 $-4{p}^{2}+420p$ . Find the revenue received when $p=90$ dollars.

Everyday Math

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 $-\dfrac{1}{150}{x}^{2}+\dfrac{1}{3}x$ . Find the fuel efficiency when $x=30 \text{mph}$ .	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 $0.06{x}^{2}+1.1x$ . Find the stopping distance when $x=40\text{mph}$ .
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.	82. Height of Projectile The height (in feet) of an object projected upward is given by the polynomial $-16{t}^{2}+60t+90$ where $t$ represents time in seconds. Find the height after $t=2.5$ seconds.
83. Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial $\dfrac{9}{5}c+32$ where $c$ represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when $c=65$ °.

Writing Exercises

84. Using your own words, explain the difference between a monomial, a binomial, and a trinomial.	85. Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.
86. Ariana thinks the sum $6{y}^{2}+5{y}^{4}$ is $11{y}^{6}$ . What is wrong with her reasoning?	87. Jonathan thinks that $\dfrac{1}{3}$ and $\dfrac{1}{x}$ are both monomials. What is wrong with his reasoning?

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	$\left({u}^{2}-6uv+5{v}^{2}\right)+\left(3{u}^{2}+2uv\right)$
Distribute.	${u}^{2}-6uv+5{v}^{2}+3{u}^{2}+2uv$
Rearrange the terms, to put like terms together.	${u}^{2}+3{u}^{2}-6uv+2uv+5{v}^{2}$
Combine like terms.	$4{u}^{2}-4uv+5{v}^{2}$

	$\left({p}^{2}+{q}^{2}\right)-\left({p}^{2}+10pq-2{q}^{2}\right)$
Distribute.	${p}^{2}+{q}^{2}-{p}^{2}-10pq+2{q}^{2}$
Rearrange the terms, to put like terms together.	${p}^{2}-{p}^{2}-10pq+{q}^{2}+2{q}^{2}$
Combine like terms.	$-10p{q}^{2}+3{q}^{2}$

	$\left({a}^{3}-{a}^{2}b\right)-\left(a{b}^{2}+{b}^{3}\right)+\left({a}^{2}b+a{b}^{2}\right)$
Distribute.	${a}^{3}-{a}^{2}b-a{b}^{2}-{b}^{3}+{a}^{2}b+a{b}^{2}$
Rearrange the terms, to put like terms together.	${a}^{3}-{a}^{2}b+{a}^{2}b-a{b}^{2}+a{b}^{2}-{b}^{3}$
Combine like terms.	${a}^{3}-{b}^{3}$

1. a) trinomial b) polynomial c) binomial d) monomial e) binomial	3. a) binomial b) trinomial c) polynomial d) trinomial e) monomial
5. a) 2 b) 4 c) 1 d) 3 e) 0	7. a) 1 b) 2 c) 3 d) 3 e) 0
9. $12{x}^{2}$	11. $6w$
13. $-5a$	15. $40x$
17. $-22b$	19. $-10a+5b$
21. $-4{a}^{2}+{b}^{2}$	21. $-4{a}^{2}+{b}^{2}$
25. ${a}^{2}b-4a-5a{b}^{2}$	27. $\text{12a}+8b$
29. $-4a-3b$	31. $-17{x}^{6}$
33. $11{y}^{2}+4y+11$	35. $-3{x}^{2}+17x-1$
37. $11{x}^{2}-5x+5$	39. $6{a}^{2}-4a-1$
41. $2{m}^{2}-7m+4$	43. $11a+3$
45. $12{s}^{2}-14s+9$	47. $3{x}^{2}-x+4$
49. ${w}^{2}+3w+4$	51. $2{p}^{3}+{p}^{2}+9p+10$
51. $2{p}^{3}+{p}^{2}+9p+10$	55. $11w-64$
57. $12c-45$	59. $10{x}^{2}-7xy+6{y}^{2}$
61. $10{m}^{2}+3mn-8{n}^{2}$	63. $-3ab+3{b}^{2}$
65. $4uv+2{v}^{2}$	67. ${p}^{3}-6{p}^{2}q+p{q}^{2}+4{q}^{3}$
69. ${x}^{3}+2{x}^{2}y-5x{y}^{2}+{y}^{3}$	71. a) 187 b) 46 c) 2
73. a) −104 b) 4 c) 40	75. 11
77. $10,800	77. $10,800
81. $58	83. 149
85. Answers will vary.	87. Answers will vary.

	${c}^{2}+7{d}^{2}-6{c}^{2}$
Combine like terms.	$-5{c}^{2}+7{d}^{2}$

	${u}^{2}v+5{u}^{2}-3{v}^{2}$
There are no like terms to combine.	${u}^{2}v+5{u}^{2}-3{v}^{2}$

Identify like terms.
Rearrange to get the like terms together.
Combine like terms.


Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

Identify Polynomials, Monomials, Binomials and Trinomials

Determine the Degree of Polynomials

Add and Subtract Monomials

Add and Subtract Polynomials

Evaluate a Polynomial for a Given Value

Key Concepts

Glossary

Practice Makes Perfect

Identify Polynomials, Monomials, Binomials, and Trinomials

Determine the Degree of Polynomials

Add and Subtract Monomials

Add and Subtract Polynomials

Evaluate a Polynomial for a Given Value

Everyday Math

Writing Exercises

Answers:

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