MX Algebra Add and Subtract Polynomials

Section 5.1Add and Subtract Polynomials

Definition

Before you get started, take this readiness quiz.

1

Simplify: \(3{x}^{2}+3x+1+8{x}^{2}+5x+5.\)
If you missed this problem, review Example 7.

\(11{x}^{2}+8x+6\)

Definition
2

Subtract: \((5n+8)-(2n-1).\)
If you missed this problem, review Example 5.

\(3n+9\)

Definition
3

Evaluate: \(4x{y}^{2}\) when \(x=-2\) and \(y=5.\)
If you missed this problem, review Example 10.

\(-200\)

Determine the Degree of 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 monomials in one variable are \(2x,5y,17z,\) and \(4{y}^{2}\) . Monomials can also have more than one variable such as \(5abc\) and \(-4{a}^{2}{b}^{3}{c}^{2}.\)

Monomial

A monomial is an algebraic expression with one term.

A monomial in one variable is a term of the form \(a{x}^{m},\) where a is a constant and m is a 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 algebraic terms 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.

Table 1
Polynomial\(y+1\)\(4{a}^{2}-7ab+2{b}^{2}\)\(4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1\)
Monomial14\(8{y}^{2}\)\(-9{x}^{3}{y}^{5}\)\(-13{a}^{3}{b}^{2}c\)
Binomial\(a+7b\)\(4{x}^{2}-{y}^{2}\)\({y}^{2}-16\)\(3{p}^{3}q-9{p}^{2}q\)
Trinomial\({x}^{2}-7x+12\)\(9{m}^{2}+2mn-8{n}^{2}\)\(6{k}^{4}-{k}^{3}+8k\)\({z}^{4}+3{z}^{2}-1\)

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.

Degree of a Polynomial

The degree of a term is the sum of the exponents of its variables.

The degree of a constant is 0.

The degree of a polynomial is the highest degree of all its terms.

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.

Let's start by looking at a monomial. The monomial \(8a{b}^{2}\) 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, \(1+2,\) is 3 so the degree is 3.

The polynomial is 8 a b squared. The exponents of the variables are 1 and 2 so the degree of the monomial is 1 plus 2 which equals 3.

Here are some additional examples.

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.

Example 1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ \(7{y}^{2}-5y+3\) ⓑ \(-2{a}^{4}{b}^{2}\) ⓒ \(3{x}^{5}-4{x}^{3}-6{x}^{2}+x-8\) ⓓ \(2y-8x{y}^{3}\) ⓔ 15

Count the terms to classify each polynomial, then add the exponents in each term to find its degree.

Table 2
PolynomialNumber of termsTypeDegree of termsDegree of polynomial
ⓐ\(7{y}^{2}-5y+3\)3Trinomial2, 1, 02
ⓑ\(-2{a}^{4}{b}^{2}\)1Monomial66
ⓒ\(3{x}^{5}-4{x}^{3}-6{x}^{2}+x-8\)5Polynomial5, 3, 2, 1, 05
ⓓ\(2y-8x{y}^{3}\)2Binomial1, 44
ⓔ151Monomial00
Try It #1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ \(-5\) ⓑ \(8{y}^{3}-7{y}^{2}-y-3\) ⓒ \(-3{x}^{2}y-5xy+9x{y}^{3}\) ⓓ \(81{m}^{2}-4{n}^{2}\) ⓔ \(-3{x}^{6}{y}^{3}z\)

ⓐ monomial, 0
ⓑ polynomial, 3 ⓒ trinomial, 4
ⓓ binomial, 2 ⓔ monomial, 10

Did you get it?
Try It #2

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ \(64{k}^{3}-8\) ⓑ \(9{m}^{3}+4{m}^{2}-2\) ⓒ \(\frac{5}{6}\) ⓓ \(8{a}^{4}-7{a}^{3}b-6{a}^{2}{b}^{2}-4a{b}^{3}+7{b}^{4}\) ⓔ \(\text{-}{p}^{4}{q}^{3}\)

ⓐ binomial, 3 ⓑ trinomial, 3 ⓒ monomial, 0 ⓓ polynomial, 4 ⓔ monomial, 7

Did you get it?

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.

Example 2

Add or subtract: ⓐ \(25{y}^{2}+15{y}^{2}\) ⓑ \(16p{q}^{3}-(-7p{q}^{3}).\)

Confirm each pair are like terms, then combine their coefficients.

ⓐ
\(\begin{array}{llll} & & & \,25{y}^{2}+15{y}^{2} \\ \text{Combine like terms.} & & & \,40{y}^{2}\end{array}\)

ⓑ
\(\begin{array}{llll} & & & \,16p{q}^{3}-(-7p{q}^{3}) \\ \text{Combine like terms.} & & & \,23p{q}^{3}\end{array}\)

Try It #3

Add or subtract: ⓐ \(12{q}^{2}+9{q}^{2}\) ⓑ \(8m{n}^{3}-(-5m{n}^{3}).\)

ⓐ \(21{q}^{2}\) ⓑ \(13m{n}^{3}\)

Did you get it?
Try It #4

Add or subtract: ⓐ \(-15{c}^{2}+8{c}^{2}\) ⓑ \(-15{y}^{2}{z}^{3}-(-5{y}^{2}{z}^{3}).\)

ⓐ \(-7{c}^{2}\) ⓑ \(-10{y}^{2}{z}^{3}\)

Did you get it?

Remember that like terms must have the same variables with the same exponents.

Example 3

Simplify: ⓐ \({a}^{2}+7{b}^{2}-6{a}^{2}\) ⓑ \({u}^{2}v+5{u}^{2}-3{v}^{2}.\)

Look for terms with matching variables and exponents before combining anything.

ⓐ

Table 3
\(\,{a}^{2}+7{b}^{2}-6{a}^{2}\)
Combine like terms.\(\,-5{a}^{2}+7{b}^{2}\)

ⓑ

Table 4
\({u}^{2}v+5{u}^{2}-3{v}^{2}\)
There are no like terms to combine.
In this case, the polynomial is unchanged.		\({u}^{2}v+5{u}^{2}-3{v}^{2}\)
Try It #5

Add: ⓐ \(8{y}^{2}+3{z}^{2}-3{y}^{2}\) ⓑ \({m}^{2}{n}^{2}-8{m}^{2}+4{n}^{2}.\)

ⓐ \(5{y}^{2}+3{z}^{2}\)
ⓑ \({m}^{2}{n}^{2}-8{m}^{2}+4{n}^{2}\)

Did you get it?
Try It #6

Add: ⓐ \(3{m}^{2}+{n}^{2}-7{m}^{2}\) ⓑ \(p{q}^{2}-6p-5{q}^{2}.\)

ⓐ \(-4{m}^{2}+{n}^{2}\)
ⓑ \(p{q}^{2}-6p-5{q}^{2}\)

Did you get it?

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.

Example 4

Find the sum: \((7{y}^{2}-2y+9)+(4{y}^{2}-8y-7).\)

Drop the parentheses, then rearrange so the like terms sit together.

Table 5
Identify like terms.\((\underset{\_\_\_\_}{\underset{\_\_\_\_}{7{y}^{2}}}-\underset{\_\_\_}{2y}+9)+(\underset{\_\_\_\_}{\underset{\_\_\_\_}{4{y}^{2}}}-\underset{\_\_\_}{8y}-7)\)
Rewrite without the parentheses,
rearranging to get the like terms together.		\(\underset{\_\_\_\_\_\_\_\_\_}{\underset{\_\_\_\_\_\_\_\_\_}{7{y}^{2}+4{y}^{2}}}-\underset{\_\_\_\_\_\_\_}{2y-8y}+9-7\)
Combine like terms.\(11{y}^{2}-10y+2\)
Try It #7

Find the sum: \((7{x}^{2}-4x+5)+({x}^{2}-7x+3).\)

\(8{x}^{2}-11x+8\)

Did you get it?
Try It #8

Find the sum: \((14{y}^{2}+6y-4)+(3{y}^{2}+8y+5).\)

\(17{y}^{2}+14y+1\)

Did you get it?

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

Example 5

Find the difference: \((9{w}^{2}-7w+5)-(2{w}^{2}-4).\)

Distribute the subtraction sign across the second polynomial before combining like terms.

Table 6
\((9{w}^{2}-7w+5)-(2{w}^{2}-4)\)
Distribute and identify like terms.\(\underset{\_\_\_\_}{\underset{\_\_\_\_}{9{w}^{2}}}-\underset{\_\_\_}{7w}+5-\underset{\_\_\_\_}{\underset{\_\_\_\_}{2{w}^{2}}}+4\)
Rearrange the terms.\(\underset{\_\_\_\_\_\_\_\_\_\_}{\underset{\_\_\_\_\_\_\_\_\_\_}{9{w}^{2}-2{w}^{2}}}-\underset{\_\_\_}{7w}+5+4\)
Combine like terms.\(7{w}^{2}-7w+9\)
Try It #9

Find the difference: \((8{x}^{2}+3x-19)-(7{x}^{2}-14).\)

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

Did you get it?
Try It #10

Find the difference: \((9{b}^{2}-5b-4)-(3{b}^{2}-5b-7).\)

\(6{b}^{2}+3\)

Did you get it?

To subtract \(a\) from \(b,\) we write it as \(b-a,\) placing the \(b\) first.

Example 6

Subtract \(({p}^{2}+10pq-2{q}^{2})\) from \(({p}^{2}+{q}^{2}).\)

Write the subtraction with \(({p}^{2}+{q}^{2})\) first, since that's the polynomial being subtracted from.

Table 7
\(({p}^{2}+{q}^{2})-({p}^{2}+10pq-2{q}^{2})\)
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.\(-10pq+3{q}^{2}\)
Try It #11

Subtract \(({a}^{2}+5ab-6{b}^{2})\) from \(({a}^{2}+{b}^{2}).\)

\(-5ab+7{b}^{2}\)

Did you get it?
Try It #12

Subtract \(({m}^{2}-7mn-3{n}^{2})\) from \(({m}^{2}+{n}^{2}).\)

\(7mn+4{n}^{2}\)

Did you get it?
Example 7

Find the sum: \(({u}^{2}-6uv+5{v}^{2})+(3{u}^{2}+2uv).\)

Drop the parentheses, then rearrange to put the like terms together.

Table 8
\(({u}^{2}-6uv+5{v}^{2})+(3{u}^{2}+2uv)\)
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.\(4{u}^{2}-4uv+5{v}^{2}\)
Try It #13

Find the sum: \((3{x}^{2}-4xy+5{y}^{2})+(2{x}^{2}-xy).\)

\(5{x}^{2}-5xy+5{y}^{2}\)

Did you get it?
Try It #14

Find the sum: \((2{x}^{2}-3xy-2{y}^{2})+(5{x}^{2}-3xy).\)

\(7{x}^{2}-6xy-2{y}^{2}\)

Did you get it?

When we add and subtract more than two polynomials, the process is the same.

Example 8

Simplify: \(({a}^{3}-{a}^{2}b)-(a{b}^{2}+{b}^{3})+({a}^{2}b+a{b}^{2}).\)

Distribute each sign carefully across all three polynomials before combining like terms.

Table 9
\(({a}^{3}-{a}^{2}b)-(a{b}^{2}+{b}^{3})+({a}^{2}b+a{b}^{2})\)
Distribute.\({a}^{3}-{a}^{2}b-a{b}^{2}-{b}^{3}+{a}^{2}b+a{b}^{2}\)
Rewrite without the parentheses,
rearranging to get the like terms together.		\({a}^{3}-{a}^{2}b+{a}^{2}b-a{b}^{2}+a{b}^{2}-{b}^{3}\)
Combine like terms.\({a}^{3}-{b}^{3}\)
Try It #15

Simplify: \(({x}^{3}-{x}^{2}y)-(x{y}^{2}+{y}^{3})+({x}^{2}y+x{y}^{2}).\)

\({x}^{3}-{y}^{3}\)

Did you get it?
Try It #16

Simplify: \(({p}^{3}-{p}^{2}q)+(p{q}^{2}+{q}^{3})-({p}^{2}q+p{q}^{2}).\)

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

Did you get it?

Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example, \(f(x)={x}^{2}+5x+6\) and \(g(x)=3x-4\) are polynomial functions, because \({x}^{2}+5x+6\) and \(3x-4\) are polynomials.

Polynomial Function

A polynomial function is a function whose range values are defined by a polynomial.

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.

Example 9

For the function \(f(x)=5{x}^{2}-8x+4\) find: ⓐ \(f(4)\) ⓑ \(f(-2)\) ⓒ \(f(0).\)

Substitute each given value for x in the formula before simplifying.

ⓐ

Table 10
The image shows the function f(x) = 5x^2 - 8x + 4.
The image shows the text 'To find f(4), substitute 4 for x.' This is a common instruction in algebra for evaluating a function at a specific point. The mathematical equation shows the evaluation of a function f(x) at x=4, represented as f(4) = 5(4)^2 - 8(4) + 4. The number 4 is highlighted in red throughout the expression.
Simplify the exponents.A mathematical equation is displayed, showing f(4) equals 5 multiplied by 16, minus 8 multiplied by 4, plus 4. It represents a function evaluation with numerical operations.
Multiply.A mathematical equation is displayed, showing 'f(4) = 80 - 32 + 4' in black text against a white background.
Simplify.A mathematical expression reads 'f(4) = 52' in black text against a white background.

ⓑ

Table 11
A mathematical equation is displayed on a white background: f(x) = 5x^2 - 8x + 4.
To find f(-2), substitute -2 for x. An algebraic equation showing f(-2) = 5(-2)^2 - 8(-2) + 4, with the number -2 highlighted in red in each instance it appears.
Simplify the exponents.Evaluating the function f(x) at x = -2: f(-2) = 5 * 4 - 8(-2) + 4. The calculation shows substituting -2 for x in the expression.
Multiply.A mathematical equation is shown, displaying f(-2) = 20 + 16 + 4, which simplifies to f(-2) = 40. This depicts a function evaluation with a numeric result.
Simplify.The image displays the mathematical expression 'f(-2) = 40' in a black serif font against a plain white background, indicating that the function f evaluated at -2 equals 40.

ⓒ

Table 12
A mathematical equation displays the function f(x) = 5x^2 - 8x + 4, a quadratic equation in standard form.
The text reads: To find f(0), substitute 0 for x. A mathematical equation is displayed, showing f(0) equals 5 multiplied by 0 squared, minus 8 multiplied by 0, plus 4. The zeros inside the parentheses are highlighted in red.
Simplify the exponents.The mathematical equation f(0) = 5 * 0 - 8(0) + 4 is shown, representing the evaluation of a function at x = 0.
Multiply.A mathematical equation is displayed on a white background, reading 'f(0) = 0 + 0 + 4' in black text.
Simplify.A mathematical expression on a white background states f(0) = 4, indicating that the function f evaluated at 0 is equal to 4.
Try It #17

For the function \(f(x)=3{x}^{2}+2x-15,\) find ⓐ \(f(3)\) ⓑ \(f(-5)\) ⓒ \(f(0).\)

ⓐ 18 ⓑ 50 ⓒ \(-15\)

Did you get it?
Try It #18

For the function \(g(x)=5{x}^{2}-x-4,\) find ⓐ \(g(-2)\) ⓑ \(g(-1)\) ⓒ \(g(0).\)

ⓐ 18 ⓑ 2 ⓒ \(-4\)

Did you get it?

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.

Example 10

The polynomial function \(h(t)=-16{t}^{2}+250\) 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.

Substitute \(t=2\) into the formula, then simplify.

Table 13
\(h(t)=-16{t}^{2}+250\)
To find \(h(2),\) substitute \(t=2.\)\(h(2)=-16{(2)}^{2}+250\)
Simplify.\(h(2)=-16·4+250\)
Simplify.\(h(2)=-64+250\)
Simplify.\(h(2)=186\)
After 2 seconds the height of the ball is 186 feet.
Try It #19

The polynomial function \(h(t)=-16{t}^{2}+150\) gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after \(t=0\) seconds (the initial height of the object).

The height is \(150\) feet.

Did you get it?
Try It #20

The polynomial function \(h(t)=-16{t}^{2}+175\) gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after \(t=3\) seconds.

The height is 31 feet.

Did you get it?

Add and Subtract Polynomial Functions

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

Addition and Subtraction of Polynomial Functions

For functions \(f(x)\) and \(g(x),\)

\[\begin{array}{l}(f+g)(x)=f(x)+g(x) \\ (f-g)(x)=f(x)-g(x)\end{array}\]

Example 11

For functions \(f(x)=3{x}^{2}-5x+7\) and \(g(x)={x}^{2}-4x-3,\) find:

ⓐ \((f+g)(x)\) ⓑ \((f+g)(3)\) ⓒ \((f-g)(x)\) ⓓ \((f-g)(-2).\)

Apply the definitions \((f+g)(x)=f(x)+g(x)\) and \((f-g)(x)=f(x)-g(x)\) before substituting the given input.

ⓐ

Table 14
The image displays the sum of two functions, f and g, as (f + g)(x) = f(x) + g(x), illustrating that the sum of functions is defined by adding their individual outputs at each point x.
The image shows the instruction 'Substitute f(x) = 3x^2 - 5x + 7 and g(x) = x^2 - 4x - 3.' The expression for f(x) is in red, and g(x) is in blue. The image shows the sum of two polynomial functions, (f+g)(x), represented as the addition of (3x² - 5x + 7) and (x² - 4x - 3).
Rewrite without the parentheses.A mathematical expression shows the sum of two functions, (f+g)(x), as 3x^2 - 5x + 7 + x^2 - 4x - 3, written in a clear, digital font on a white background.
Put like terms together.A mathematical equation shows the sum of two functions, (f + g)(x), expanded as 3x^2 + x^2 - 5x - 4x + 7 - 3, likely as an intermediate step to simplification.
Combine like terms.The image displays the sum of two functions, (f + g)(x), expressed as the quadratic equation 4x^2 - 9x + 4.

ⓑ In part (a) we found \((f+g)(x)\) and now are asked to find \((f+g)(3).\)

Table 15
\((f+g)(x)=4{x}^{2}-9x+4\)
To find \((f+g)(3),\) substitute \(x=3.\)\((f+g)(3)=4{(3)}^{2}-9·3+4\)
\((f+g)(3)=4·9-9·3+4\)
\((f+g)(3)=36-27+4\)

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.

Table 16
Find \(f(3).\)The image displays the quadratic function f(x) = 3x^2 - 5x + 7 in black text against a white background.
The image shows a mathematical equation: f(3) = 3(3)^2 - 5(3) + 7. The number 3 is highlighted in red, indicating its substitution into the function.
A mathematical expression 'f(3) = 19' is displayed in black text on a white background, representing a function f evaluated at 3 equals 19.
Find \(g(3).\)The image displays the quadratic function g(x) = x^2 - 4x - 3 in black text against a white background, representing a mathematical equation.
A mathematical equation is displayed, showing a function g evaluated at 3. The equation reads: g(3) = 3^2 - 4(3) - 3, with the number '3' highlighted in red wherever it appears in the expression.
The image shows the mathematical equation g(3) = -6 on a white background. The function g evaluated at 3 equals negative 6.
Find \((f+g)(3).\)A mathematical equation shows the sum of two functions, f and g, applied to x, is equal to the sum of each function applied individually to x: (f + g)(x) = f(x) + g(x).
The image shows the sum of two functions, f and g, evaluated at 3, which equals the sum of each function evaluated at 3: (f + g)(3) = f(3) + g(3).
\(\,\)The image shows the text 'Substitute f(3) = 19 and g(3) = -6.' The number 19 is highlighted in red, and the number -6 is highlighted in teal. A mathematical equation shows (f + g)(3) = 19 + (-6), representing the sum of two functions f and g evaluated at 3, equaling the addition of 19 and -6.
A mathematical equation showing the sum of two functions f and g evaluated at 3 equals 13, written as (f + g)(3) = 13.

ⓒ

Table 17
The image displays the definition of the difference between two functions, stating that (f-g)(x) is equal to f(x) - g(x).
Two functions, f(x) = 3x^2 - 5x + 7 (red) and g(x) = x^2 - 4x - 3 (blue), are presented for substitution in a mathematical problem. Mathematical expression for (f-g)(x) where f(x) = 3x^2 - 5x + 7 (red) and g(x) = x^2 - 4x - 3 (blue), illustrating polynomial subtraction.
Rewrite without the parentheses.The equation shows the subtraction of two functions, (f-g)(x), which equals 3x^2 - 5x + 7 - x^2 + 4x + 3. This equation represents a polynomial expression with multiple terms.
Put like terms together.The image shows the mathematical expression (f - g)(x) = 3x^2 - x^2 - 5x + 4x + 7 + 3, representing the subtraction of two functions, f(x) and g(x).
Combine like terms.The image displays the subtraction of two functions, represented as (f - g)(x), which equals the quadratic expression 2x^2 - x + 10.

ⓓ

Table 18
The image demonstrates the process of evaluating the function (f-g)(x) = 2x^2 - x + 10 at x = -2, showing step-by-step substitution and calculation to arrive at the result (f-g)(-2) = 20.
Try It #21

For functions \(f(x)=2{x}^{2}-4x+3\) and \(g(x)={x}^{2}-2x-6,\) find: ⓐ \((f+g)(x)\) ⓑ \((f+g)(3)\) ⓒ \((f-g)(x)\) ⓓ \((f-g)(-2).\)

ⓐ \((f+g)(x)=3{x}^{2}-6x-3\) ⓑ \((f+g)(3)=6\)
ⓒ \((f-g)(x)={x}^{2}-2x+9\)
ⓓ \((f-g)(-2)=17\)

Did you get it?
Try It #22

For functions \(f(x)=5{x}^{2}-4x-1\) and \(g(x)={x}^{2}+3x+8,\) find ⓐ \((f+g)(x)\) ⓑ \((f+g)(3)\) ⓒ \((f-g)(x)\) ⓓ \((f-g)(-2).\)

ⓐ \((f+g)(x)=6{x}^{2}-x+7\) ⓑ \((f+g)(3)=58\)
ⓒ \((f-g)(x)=4{x}^{2}-7x-9\)
ⓓ \((f-g)(-2)=21\)

Did you get it?
Media

Access this online resource for additional instruction and practice with adding and subtracting polynomials.

Key Concepts

Section Exercises

Practice Makes Perfect

Determine the Type of Polynomials

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

4


ⓐ \(47{x}^{5}-17{x}^{2}{y}^{3}+{y}^{2}\)
ⓑ \(5{c}^{3}+11{c}^{2}-c-8\)
ⓒ \(\frac{5}{9}ab+\frac{1}{3}b\)
ⓓ 4
ⓔ \(4pq+17\)

ⓐ trinomial, 5 ⓑ polynomial, 3 ⓒ binomial, 2 ⓓ monomial, 0
ⓔ binomial, 2

5


ⓐ \({x}^{2}-{y}^{2}\)
ⓑ \(-13{c}^{4}\)
ⓒ \({a}^{2}+2ab-7{b}^{2}\)
ⓓ \(4{x}^{2}{y}^{2}-3xy+8\)
ⓔ 19

6


ⓐ \(8y-5x\)
ⓑ \({y}^{2}-5yz-6{z}^{2}\)
ⓒ \({y}^{3}-8{y}^{2}+2y-16\)
ⓓ \(81a{b}^{4}-24{a}^{2}{b}^{2}+3b\)
ⓔ \(-18\)

ⓐ binomial, 1ⓑ trinomial, 2
ⓒ polynomial, 3ⓓ trinomial, 5
ⓔ monomial, 0

7


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

8


ⓐ \(5{a}^{2}+12ab-7{b}^{2}\)
ⓑ \(18x{y}^{2}z\)
ⓒ \(5x+2\)
ⓓ \({y}^{3}-8{y}^{2}+2y-16\)
ⓔ \(-24\)

ⓐ \(\text{trinomial, }2\) ⓑ \(\text{monomial, }4\) ⓒ \(\text{binomial, }1\) ⓓ \(\text{polynomial, }3\)
ⓔ \(\text{monomial, }0\)

9


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

10


ⓐ \(14s-29t\)
ⓑ \({z}^{2}-5z-6\)
ⓒ \({y}^{3}-8{y}^{2}z+2y{z}^{2}-16{z}^{3}\)
ⓓ \(23a{b}^{2}-14\)
ⓔ \(-3\)

ⓐ \(\text{binomial, }1\) ⓑ \(\text{trinomial, }2\) ⓒ \(\text{polynomial, }3\) ⓓ \(\text{binomial, }3\)
ⓔ \(\text{monomial, }0\)

11


ⓐ \(15xy\)
ⓑ 15
ⓒ \(6{x}^{2}-3xy+4x-2y+{y}^{2}\)
ⓓ \(10p-9q\)
ⓔ \({m}^{4}+4{m}^{3}+6{m}^{2}+4m+1\)

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

12


ⓐ \({\,\text{7x}}^{\text{2}}+5{x}^{2}\)
ⓑ \(\,\text{4a}-9a\)

ⓐ \({\text{12x}}^{\text{2}}\) ⓑ \(\text{-}\,\text{5a}\)

13


ⓐ \({\,\text{4y}}^{\text{3}}+6{y}^{3}\)
ⓑ \(\text{-}y-5y\)

14


ⓐ \(-12w+18w\)
ⓑ \(7{x}^{2}y-(-12{x}^{2}y)\)

ⓐ \(\text{6}w\) ⓑ \(19{x}^{2}y\)

15


ⓐ \(-3m+9m\)
ⓑ \(15y{z}^{2}-(-8y{z}^{2})\)

16

\({\,\text{7x}}^{\text{2}}+5{x}^{2}+\,\text{4a}-9a\)

\({\text{12x}}^{\text{2}}-\,\text{5a}\)

17

\({\,\text{4y}}^{\text{3}}+6{y}^{3}-y-5y\)

18

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

\(6w+19{x}^{2}y\)

19

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

20


ⓐ \(-5b-17b\)
ⓑ \(3xy-(-8xy)+5xy\)

ⓐ \(-22b\) ⓑ \(16xy\)

21


ⓐ \(-10x-35x\)
ⓑ \(17m{n}^{2}-(-9m{n}^{2})+3m{n}^{2}\)

22


ⓐ \(\,\text{12}a+5b-22a\)
ⓑ \(p{q}^{2}-4p-3{q}^{2}\)

ⓐ \(-10a+5b\)
ⓑ \(p{q}^{2}-4p-3{q}^{2}\)

23


ⓐ \(\,\text{14x}-3y-13x\)
ⓑ \({a}^{2}b-4a-5a{b}^{2}\)

24


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

ⓐ \(-4{a}^{2}+{b}^{2}\)
ⓑ \({x}^{2}y-3x+7x{y}^{2}\)

25


ⓐ \(5{u}^{2}+4{v}^{2}-6{u}^{2}\)
ⓑ \(\,\text{12a}+8b\)

26


ⓐ \(x{y}^{2}-5x-5{y}^{2}\)
ⓑ \(\,\text{19y}+5z\)

ⓐ \(x{y}^{2}-5x-5{y}^{2}\)
ⓑ \(19y+5z\)

27

\(\text{12}a+5b-22a+p{q}^{2}-4p-3{q}^{2}\)

28

\(\text{14x}-3y-13x+{a}^{2}b-4a-5a{b}^{2}\)

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

29

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

30

\(5{u}^{2}+4{v}^{2}-6{u}^{2}+\,\text{12a}+8b\)

\(\text{-}{u}^{2}+4{v}^{2}+\,\text{12a}+8b\)

31

\(x{y}^{2}-5x-5{y}^{2}+\,\text{19y}+5z\)

32

Add: \(4a,-3b,-8a\)

\(\text{-}\text{4a}-3b\)

33

Add: \(\,\text{4x},3y,-3x\)

34

Subtract \(5{x}^{6}\) from \(-12{x}^{6}\)

\(-17{x}^{6}\)

35

Subtract \(2{p}^{4}\) from \(-7{p}^{4}\)

In the following exercises, add the polynomials.

36

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

\(11{y}^{2}+4y+11\)

37

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

38

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

\(-3{x}^{2}+17x-1\)

39

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

40

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

\(11{x}^{2}-5x+5\)

41

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

42

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

\(6{a}^{2}-4a-1\)

43

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

In the following exercises, subtract the polynomials.

44

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

\(2{m}^{2}-7m+4\)

45

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

46

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

\(11a+3\)

47

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

48

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

\(12{s}^{2}-16s+9\)

49

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

In the following exercises, subtract the polynomials.

50

Subtract \((9{x}^{2}+2)\) from \((12{x}^{2}-x+6)\)

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

51

Subtract \((5{y}^{2}-y+12)\) from \((10{y}^{2}-8y-20)\)

52

Subtract \((7{w}^{2}-4w+2)\) from \((8{w}^{2}-w+6)\)

\({w}^{2}+3w+4\)

53

Subtract \((5{x}^{2}-x+12)\) from \((9{x}^{2}-6x-20)\)

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

54

Find the difference of \(({w}^{2}+w-42)\) and \(({w}^{2}-10w+24)\)

\(11w-66\)

55

Find the difference of \(({z}^{2}-3z-18)\) and \(({z}^{2}+5z-20)\)

In the following exercises, add the polynomials.

56

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

\(10{x}^{2}-7xy+6{y}^{2}\)

57

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

58

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

\(10{m}^{2}+3mn-8{n}^{2}\)

59

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

In the following exercises, add or subtract the polynomials.

60

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

\(-3ab+3{b}^{2}\)

61

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

62

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

\({p}^{3}-6{p}^{2}q+p{q}^{2}+4{q}^{3}\)

63

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

64

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

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

65

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

Evaluate a Polynomial Function for a Given Value

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

66

For the function \(f(x)=8{x}^{2}-3x+2,\) find:
ⓐ \(f(5)\) ⓑ \(f(-2)\) ⓒ \(f(0)\)

ⓐ 187 ⓑ 40 ⓒ 2

67

For the function \(f(x)=5{x}^{2}-x-7,\) find:
ⓐ \(f(-4)\) ⓑ \(f(1)\) ⓒ \(f(0)\)

68

For the function \(g(x)=4-36x,\) find:
ⓐ \(g(3)\) ⓑ \(g(0)\) ⓒ \(g(-1)\)

ⓐ \(-104\) ⓑ 4 ⓒ 40

69

For the function \(g(x)=16-36{x}^{2},\) find:
ⓐ \(g(-1)\) ⓑ \(g(0)\) ⓒ \(g(2)\)

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

70

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.

71

A girl drops a ball off a 200-foot cliff into the ocean. The polynomial \(h(t)=-16{t}^{2}+200\) gives the height of the ball, in feet, t seconds after it is dropped. Find the height after \(t=3\) seconds.

72

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 \(R(p)=-4{p}^{2}+420p.\) Find the revenue received when \(p=60\) dollars.

The revenue is $10,800.

73

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 \(R(p)=-4{p}^{2}+420p.\) Find the revenue received when \(p=90\) dollars.

74

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

75

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

In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

76

\(f(x)=2{x}^{2}-4x+1\) and \(g(x)=5{x}^{2}+8x+3\)

ⓐ \((f+g)(x)=7{x}^{2}+4x+4\) ⓑ \((f+g)(2)=40\)
ⓒ \((f-g)(x)=-3{x}^{2}-12x-2\)
ⓓ \((f-g)(-3)=7\)

77

\(f(x)=4{x}^{2}-7x+3\) and \(g(x)=4{x}^{2}+2x-1\)

78

\(f(x)=3{x}^{3}-{x}^{2}-2x+3\) and \(g(x)=3{x}^{3}-7x\)


ⓐ \((f+g)(x)=6{x}^{3}-{x}^{2}-9x+3\)
ⓑ \((f+g)(2)=29\)
ⓒ \((f-g)(x)=\text{-}{x}^{2}+5x+3\)
ⓓ \((f-g)(-3)=-21\)

79

\(f(x)=5{x}^{3}-{x}^{2}+3x+4\) and \(g(x)=8{x}^{3}-1\)

Writing Exercises

80

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

Answers will vary.

81

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

82

Ariana thinks the sum \(6{y}^{2}+5{y}^{4}\) is \(11{y}^{6}.\) What is wrong with her reasoning?

Answers will vary.

83

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.

The figure shows a table with six rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “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”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.

ⓑ 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. Whom 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.

Glossary

binomial
A binomial is a polynomial with exactly two terms.
degree of a constant
The degree of any constant is 0.
degree of a polynomial
The degree of a polynomial is the highest degree of all its terms.
degree of a term
The degree of a term is the sum of the exponents of its variables.
monomial
A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form \(a{x}^{m},\) where a is a constant and m is a whole number.
polynomial
A monomial or two or more monomials combined by addition or subtraction is a polynomial.
standard form of a polynomial
A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.
trinomial
A trinomial is a polynomial with exactly three terms.
polynomial function
A polynomial function is a function whose range values are defined by a polynomial.