Add and Subtract Polynomials

We have learned that a term is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. 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 in one variable. Some examples of monomial in one variable are. Monomials can also have more than one variable such as and

−4 a^{2} b^{3} c^{2} .

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.

| {: valign=”top”}{: .unnumbered summary=”This table has five columns and four rows. The first row is for polynomials and lists three examples: y plus 1, 4 a squared minus 7 a b plus 2 b squared, 4 x to the fourth power plus x cubed plus 8 x squared minus 9 x plus 1. The second row is for monomials and lists four examples: 14, 8 y squared, minus 9 x cubed y to the fifth power, and negative 12 a cubed b squared c. The third row is for binomials and lists four examples: a plus 7 b, 4 x square minus y squared, y squared minus 16, and 3 p cubed q minus 9 p squared q. The fourth row is for trinomials and list four examples: x squared minus 7 x plus 12, 9 m squared plus 2 mn minus 8 n squared, 6 k to the fourth power minus k cubed plus 8 k, and z to the fourth power plus 3z squared minus 1.” data-label=””}

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.

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents,

1 + 2,

Monomial examples: 14 has degree 0, 8 a b squared has degree 3, negative 9 x cubed y to the fifth power has degree 8, negative 13 a has degree 1. Binomial examples: The terms in h plus 7 have degree 1 and 0 so the degree of the whole polynomial is 1. The terms in 7 b squared minus 3 b have degree 2 and 1 so the degree of the whole polynomial is 2. The terms in z squared y squared minus 25 have degree 4 and 0 so the degree of the whole polynomial is 4. The terms in 4 n cubed minus 8 n squared have degree 3 and 2 so the degree of the whole polynomial is 3. Trinomial examples: The terms in x squared minus 12 x plus 27 have degree 2, 1 and 0 so the degree of the whole polynomial is 2. The terms in 9 a squared plus 6 a b plus b squared have degree 2, 2, and 2 so the degree of the whole polynomial is 2. The terms in 6 m to the fourth power minus m cubed n squared plus 8 m n to the fifth power have degree 4, 5, and 6 so the degree of the whole polynomial is 6. The terms in z to the fourth power plus 3 z squared minus 1 have degree 4, 2, and 0 so the degree of the whole polynomial is 4. Polynomial examples: The terms in y minus 1 have degree 1 and 0 so the degree of the whole polynomial is 1. The terms in 3 y squared minus 2 y minus 5 have degree 2, 1, 0 so the degree of the whole polynomial is 2. The terms in 4 x to the fourth power plus x cubed plus eight x squared minus 9 x plus 1 have degree 4, 3, 2, 1, and 0 so the degree of the whole polynomial is 4.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial. Get in the habit of writing the term with the highest degree first.

Add and Subtract Polynomials

We have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficients.

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Be careful with the signs as you distribute while subtracting the polynomials in the next example.

Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example,

f (x) = x^{2} + 5 x + 6

In Graphs and Functions, where we first introduced functions, we learned that evaluating a function means to find the value of

f (x)

for a given value of x. To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.

The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.

Add and Subtract Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

For functions $f (x) = 3 x^{2} - 5 x + 7$

and $g (x) = x^{2} - 4 x - 3,$

find:

ⓐ $(f + g) (x)$

ⓑ $(f + g) (3)$

ⓓ $(f - g) (−2) .$

ⓐ* * *

| | | {: valign=”top”}| | | {: valign=”top”}| Rewrite without the parentheses. | | {: valign=”top”}| Put like terms together. | | {: valign=”top”}| Combine like terms. | | {: valign=”top”}{: .unnumbered .unstyled summary=”The figure shows a series of equations in function notation. The first equation is the formula f plus g of x equals f of x plus g of x. Substituting f of x equals 3 x squared minus 5 x plus 7 and g of x equals x squared minus 4 x minus 3 into the formula results in the equation f plus g of x equals the quantity 3 x squared minus 5 x plus 7 in parentheses plus the quantity x squared minus 4 x minus 3 in parentheses. Rewriting without parentheses is the equation f plus g of x equals 3 x squared minus 5 x plus 7 plus x squared minus 4 x minus 3. Putting like terms together results in the equation f plus g of x equals 3 x squared plus x squared minus 5 x minus 4 x plus 7 minus 3. Combining like terms results in the fully simplified function equation f plus g of x equals 4 x squared minus 9 x plus 4.” data-label=””}

ⓑ In part (a) we found $(f + g) (x)$

and now are asked to find $(f + g) (3) .$

\begin{matrix} (f + g) (x) = 4 x^{2} - 9 x + 4 \\ To find (f + g) (3), substitute x = 3 . & (f + g) (3) = 4 {(3)}^{2} - 9 \cdot 3 + 4 \\ (f + g) (3) = 4 \cdot 9 - 9 \cdot 3 + 4 \\ (f + g) (3) = 36 - 27 + 4 \end{matrix}

Notice that we could have found $(f + g) (3)$

by first finding the values of $f (3)$

and $g (3)$

separately and then adding the results.

Find

f (3) .


{: valign=”top”}
{: valign=”top”}
{: valign=”top”}	Find $g (3) .$


{: valign=”top”}
{: valign=”top”}
{: valign=”top”}	Find $(f + g) (3) .$


{: valign=”top”}
{: valign=”top”}

| | {: valign=”top”}| | | {: valign=”top”}{: .unnumbered .unstyled summary=”The figure shows a series of calculations to verify the last result. The first calculation is labeled Find f of 3. The first equation is the function equation f of x equals 3 x squared minus 5 x plus 7. The second equation is f of 3 equals 3 times 3 squared minus 5 times 3 plus 7, where the 3’s are all emphasized. The third equation is f of 3 equals 19. The second calculation is labeled Find g of 3. The first equation is the function equation g of x equals x squared minus 4 x minus 3. The second equation is g of 3 equals 3 squared minus 4 times 3 minus 3. The 3’s that replaced the x’s are emphasized. The next equation is g of 3 equals negative 6. The last calculation is labeled Find f plus g of 3. The first equation is the formula f plus g of x equals f of x plus g of x. The second equation is f plus g of 3 equals f of 3 plus g of 3. Substituting f of 3 equals 19 and g of 3 equals negative 6 we get the equation f plus g of 3 equals 19 minus 6. The last equation is f plus g of 3 equals 13.” data-label=””}

ⓒ* * *

| | | {: valign=”top”}| | | {: valign=”top”}| Rewrite without the parentheses. | | {: valign=”top”}| Put like terms together. | | {: valign=”top”}| Combine like terms. | | {: valign=”top”}{: .unnumbered .unstyled summary=”The figure shows a series of equations in function notation. The first equation is the formula f minus g of x equals f of x minus g of x. Substituting f of x equals 3 x squared minus 5 x plus 7 and g of x equals x squared minus 4 x minus 3 into the formula results in the equation f minus g of x equals the quantity 3 x squared minus 5 x plus 7 in parentheses minus the quantity x squared minus 4 x minus 3 in parentheses. Rewriting without parentheses is the equation f minus g of x equals 3 x squared minus 5 x plus 7 minus x squared plus 4 x plus 3. Putting like terms together results in the equation f minus g of x equals 3 x squared minus x squared minus 5 x plus 4 x plus 7 plus 3. Combining like terms results in the fully simplified function equation f minus g of x equals 2 x squared minus x plus 10.” data-label=””}

ⓓ* * *

| | {: valign=”top”}{: .unnumbered .unstyled summary=”The figure shows a series of equations in function notation. The first equation is the equation f minus g of x equals 2 x squared minus x plus 10. To find f minus g of negative 2 substitute x equals negative 2. The second equation is f minus g of negative 2 equals 2 times negative 2 squared minus negative 2 plus 10. The next equation is f minus g of negative 2 equals 2 times 4 minus negative 2 plus 10. The next equation is f minus g of negative 2 equals 20.” data-label=””}

Key Concepts

Practice Makes Perfect

Determine the Type of Polynomials

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

ⓐ $47 x^{5} - 17 x^{2} y^{3} + y^{2}$

ⓑ $5 c^{3} + 11 c^{2} - c - 8$

ⓓ 4* * *

ⓔ $4 p q + 17$

ⓐ trinomial, 5 ⓑ polynomial, 3 ⓒ binomial, 1 ⓓ monomial, 1* * *

ⓔ binomial, 1

ⓐ $x^{2} - y^{2}$

ⓑ $−13 c^{4}$

ⓓ $4 x^{2} y^{2} - 3 x y + 8$

ⓔ 19

ⓐ $8 y - 5 x$

ⓑ $y^{2} - 5 y z - 6 z^{2}$

ⓓ $81 a b^{4} - 24 a^{2} b^{2} + 3 b$

ⓔ $−18$

ⓐ binomial ⓑ trinomial* * *

ⓔ monomial

ⓐ $11 y^{2}$

ⓑ $−73$

ⓓ $4 y^{2} + 17 z^{2}$

ⓔ $5 c^{3} + 11 c^{2} - c - 8$

ⓐ $5 a^{2} + 12 a b - 7 b^{2}$

ⓑ $18 x y^{2} z$

ⓓ $y^{3} - 8 y^{2} + 2 y - 16$

ⓔ $−24$

ⓐ $2^{0}$

ⓑ $3^{0}$

ⓓ $3^{0}$

ⓔ $0^{0}$

ⓐ $9 y^{3} - 10 y^{2} + 2 y - 6$

ⓑ $−12 p^{3} q$

ⓓ $20 x^{2} y^{2} - 10 a^{2} b^{2} + 30$

ⓔ 17

ⓐ $14 s - 29 t$

ⓑ $z^{2} - 5 z - 6$

ⓓ $23 a b^{2} - 14$

ⓔ $−3$

ⓐ $1^{0}$

ⓑ $2^{0}$

ⓓ $3^{0}$

ⓔ $0^{0}$

ⓐ $15 x y^{}$

ⓑ 15* * *

ⓓ $10 p - 9 q$

ⓔ $m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1$

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

ⓐ ${7x}^{2} + 5 x^{2}$

ⓑ $4a - 9 a$

ⓐ ${12x}^{2}$

ⓑ $− 5a$

ⓐ ${4y}^{3} + 6 y^{3}$

ⓑ $− y - 5 y$

ⓐ $−12 w + 18 w$

ⓑ $7 x^{2} y - (−12 x^{2} y)$

ⓐ $6 w$

ⓑ $19 x^{2} y$

ⓐ $−3 m + 9 m$

ⓑ $15 y z^{2} - (−8 y z^{2})$

{7x}^{2} + 5 x^{2} + 4a - 9 a

{12x}^{2} - 5a

{4y}^{3} + 6 y^{3} - y - 5 y

−12 w + 18 w + 7 x^{2} y - (−12 x^{2} y)

6 w + 19 x^{2} y

−3 m + 9 m + 15 y z^{2} - (−8 y z^{2})

ⓐ $−5 b - 17 b$

ⓑ $3 x y - (−8 x y) + 5 x y$

ⓐ $−22 b$

ⓑ $16 x y$

ⓐ $−10 x - 35 x$

ⓑ $17 m n^{2} - (−9 m n^{2}) + 3 m n^{2}$

ⓐ $12 a + 5 b - 22 a$

ⓑ $p q^{2} - 4 p - 3 q^{2}$

ⓐ $−10 a + 5 b$

ⓑ $p q^{2} - 4 p - 3 q^{2}$

ⓐ $14x - 3 y - 13 x$

ⓑ $a^{2} b - 4 a - 5 a b^{2}$

ⓐ $2 a^{2} + b^{2} - 6 a^{2}$

ⓑ $x^{2} y - 3 x + 7 x y^{2}$

ⓐ $−4 a^{2} + b^{2}$

ⓑ $x^{2} y - 3 x + 7 x y^{2}$

ⓐ $5 u^{2} + 4 v^{2} - 6 u^{2}$

ⓑ $12a + 8 b$

ⓐ $x y^{2} - 5 x - 5 y^{2}$

ⓑ $19y + 5 z$

ⓐ $x y^{2} - 5 x - 5 y^{2}$

ⓑ $19 y + 5 z$

12 a + 5 b - 22 a + p q^{2} - 4 p - 3 q^{2}

14x - 3 y - 13 x + a^{2} b - 4 a - 5 a b^{2}

x - 3 y + a^{2} b - 4 a - 5 a b^{2}

2 a^{2} + b^{2} - 6 a^{2} + x^{2} y - 3 x + 7 x y^{2}

5 u^{2} + 4 v^{2} - 6 u^{2} + 12a + 8 b

− u^{2} + 4 v^{2} + 12a + 8 b

x y^{2} - 5 x - 5 y^{2} + 19y + 5 z

Add: $4 a, −3 b, −8 a$

− 4a - 3 b

Add: $4x, 3 y, −3 x$

Subtract $5 x^{6}$

from $−12 x^{6}$

−7 x^{6}

Subtract $2 p^{4}$

from $−7 p^{4}$

In the following exercises, add the polynomials.

(5 y^{2} + 12 y + 4) + (6 y^{2} - 8 y + 7)

11 y^{2} + 4 y + 11

(4 y^{2} + 10 y + 3) + (8 y^{2} - 6 y + 5)

(x^{2} + 6 x + 8) + (−4 x^{2} + 11 x - 9)

−3 x^{2} + 17 x - 1

(y^{2} + 9 y + 4) + (−2 y^{2} - 5 y - 1)

(8 x^{2} - 5 x + 2) + (3 x^{2} + 3)

11 x^{2} - 5 x + 5

(7 x^{2} - 9 x + 2) + (6 x^{2} - 4)

(5 a^{2} + 8) + (a^{2} - 4 a - 9)

6 a^{2} - 4 a - 1

(p^{2} - 6 p - 18) + (2 p^{2} + 11)

In the following exercises, subtract the polynomials.

(4 m^{2} - 6 m - 3) - (2 m^{2} + m - 7)

2 m^{2} - 7 m + 4

(3 b^{2} - 4 b + 1) - (5 b^{2} - b - 2)

(a^{2} + 8 a + 5) - (a^{2} - 3 a + 2)

5 a + 3

(b^{2} - 7 b + 5) - (b^{2} - 2 b + 9)

(12 s^{2} - 15 s) - (s - 9)

12 s^{2} - 14 s + 9

(10 r^{2} - 20 r) - (r - 8)

In the following exercises, subtract the polynomials.

Subtract $(9 x^{2} + 2)$

from $(12 x^{2} - x + 6)$

3 x^{2} - x + 4

Subtract $(5 y^{2} - y + 12)$

from $(10 y^{2} - 8 y - 20)$

Subtract $(7 w^{2} - 4 w + 2)$

from $(8 w^{2} - w + 6)$

w^{2} + 3 w + 4

Subtract $(5 x^{2} - x + 12)$

from $(9 x^{2} - 6 x - 20)$

In the following exercises, find the difference of the polynomials.

Find the difference of $(w^{2} + w - 42)$

and $(w^{2} - 10 w + 24)$

11 w - 64

Find the difference of $(z^{2} - 3 z - 18)$

and $(z^{2} + 5 z - 20)$

In the following exercises, add the polynomials.

(7 x^{2} - 2 x y + 6 y^{2}) + (3 x^{2} - 5 x y)

10 x^{2} - 7 x y + 6 y^{2}

(−5 x^{2} - 4 x y - 3 y^{2}) + (2 x^{2} - 7 x y)

(7 m^{2} + m n - 8 n^{2}) + (3 m^{2} + 2 m n)

10 m^{2} + 3 m n - 8 n^{2}

(2 r^{2} - 3 r s - 2 s^{2}) + (5 r^{2} - 3 r s)

In the following exercises, add or subtract the polynomials.

(a^{2} - b^{2}) - (a^{2} + 3 a b - 4 b^{2})

−3 a b + 3 b^{2}

(m^{2} + 2 n^{2}) - (m^{2} - 8 m n - n^{2})

(p^{3} - 3 p^{2} q) + (2 p q^{2} + 4 q^{3}) - (3 p^{2} q + p q^{2})

p^{3} - 6 p^{2} q + p q^{2} + 4 q^{3}

(a^{3} - 2 a^{2} b) + (a b^{2} + b^{3}) - (3 a^{2} b + 4 a b^{2})

(x^{3} - x^{2} y) - (4 x y^{2} - y^{3}) + (3 x^{2} y - x y^{2})

x^{3} + 2 x^{2} y - 5 x y^{2} + y^{3}

(x^{3} - 2 x^{2} y) - (x y^{2} - 3 y^{3}) - (x^{2} y - 4 x y^{2})

Evaluate a Polynomial Function for a Given Value

In the following exercises, find the function values for each polynomial function.

For the function $f (x) = 8 x^{2} - 3 x + 2,$

find:* * *

ⓐ $f (5)$

ⓑ $f (−2)$

ⓐ 187 ⓑ 40 ⓒ 2

For the function $f (x) = 5 x^{2} - x - 7,$

find:* * *

ⓐ $f (−4)$

ⓑ $f (1)$

For the function $g (x) = 4 - 36 x,$

find:* * *

ⓐ $g (3)$

ⓑ $g (0)$

ⓐ $−104$

ⓑ 4 ⓒ 40

For the function $g (x) = 16 - 36 x^{2},$

find:* * *

ⓐ $g (−1)$

ⓑ $g (0)$

In the following exercises, find the height for each polynomial function.

A painter drops a brush from a platform 75 feet high. The polynomial function $h (t) = −16 t^{2} + 75$

gives the height of the brush t seconds after it was dropped. Find the height after $t = 2$

seconds.

The height is 11 feet.

A girl is throwing a ball off the cliff into the ocean. The polynomial $h (t) = −16 t^{2} + 200$

gives the height of a ball t seconds after it is dropped from a 250-foot tall cliff. Find the height after $t = 3$

seconds.

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 function $R (p) = −4 p^{2} + 420 p .$

Find the revenue received when $p = 60$

dollars.

The revenue is $10,800.

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 $R (p) = −4 p^{2} + 420 p .$

Find the revenue received when $p = 90$

dollars.

The polynomial $C (x) = 6 x^{2} + 90 x$

gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 6 feet. Find the cost of producing a box with $x = 4$

feet.

The cost is $456.

The polynomial $C (x) = 6 x^{2} + 90 x$

gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 4 feet. Find the cost of producing a box with $x = 6$

feet.

Add and Subtract Polynomial Functions

f (x) = 2 x^{2} - 4 x + 1

and $g (x) = 5 x^{2} + 8 x + 3$

ⓐ $(f + g) (x) = 7 x^{2} + 4 x + 4$

ⓑ $(f + g) (2) = 40$

ⓓ $(f - g) (−3) = 7$

f (x) = 4 x^{2} - 7 x + 3

and $g (x) = 4 x^{2} + 2 x - 1$

f (x) = 3 x^{3} - x^{2} - 2 x + 3

and $g (x) = 3 x^{3} - 7 x$

ⓐ $(f + g) (x) = 6 x^{3} - x^{2} - 9 x + 3$

ⓑ $(f + g) (2) = 29$

ⓓ $(f - g) (−3) = −21$

f (x) = 5 x^{3} - x^{2} + 3 x + 4

and $g (x) = 8 x^{3} - 1$

Writing Exercises

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

Answers will vary.

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

Ariana thinks the sum $6 y^{2} + 5 y^{4}$

is $11 y^{6} .$

What is wrong with her reasoning?

Answers will vary.

Is every trinomial a second degree polynomial? If not, give an example.

Self Check

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

ⓑ 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.

	Polynomial	Number of terms	Type	Degree of terms	Degree of polynomial
{: valign=”top”}	———-
ⓐ	$7 y^{2} - 5 y + 3$

Determine the Degree of Polynomials