Learning Objectives

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

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

Try it

Before you get started, take this readiness quiz:

1) Simplify: $8x+3x$.
2) Subtract: $(5n+8)-(2n-1)$
3) Write in expanded form: ${a}^{5}$.

Identify Polynomials, Monomials, binomial and Trinomials

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 $11{z}^{7}$.

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

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

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.

Polynomial Monomial Binomial Trinomial $b+1$ $4y^2-7y+2$ $4x^4+x^3+8x^2-9x+1$ $14$ $8y^2$ $-9x^3y^5$ $-13$ $a+7$ $4b-5$ $y^2-16$ $3x^3-9x^2$ $x^2-7x+12$ $9y^2+2y-8$ $6m^4-m^3+8m$ $z^4+3z^2-1$

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.

Example 5.1.1

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

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

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

4) 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. $-5{x}^{6}$

Solution

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

5) 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. $\frac{5}{6}$
d. $8{x}^{4}-7{x}^{2}-6x-5$
e. ${−}{n}^{4}$

Solution

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

Determine the Degree of 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 $0$—it has no variable.

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.

 Monomial Degree \begin{align*}14\\[2ex]{\color{blue}{0}}\;\end{align*} \begin{align*}8y^2\\[2ex]{\color{blue}{2}}\;\end{align*} \begin{align*}-9x^3y^5\\[2ex]{\color{blue}{8}}\;\;\;\;\end{align*} \begin{align*}-13a\\[2ex]{\color{blue}{1}}\;\end{align*} Binomial Degree of each term Degree of polynomial \begin{align*}a+7\\[2ex]{\color{blue}{1\;\;\;\;0}}\\[2ex]\;{\color{red}{1}}\;\;\;\end{align*} \begin{align*}4b^2-5b\\[2ex]{\color{blue}{2\;\;\;\;\;1}}\;\\[2ex]\;{\color{red}{2}}\;\;\;\;\end{align*} \begin{align*}x^2y^2-16\\[2ex]{\color{blue}{4\;\;\;\;\;\;\;0}}\;\\[2ex]\;{\color{red}{4}}\;\;\;\;\;\end{align*} \begin{align*}3n^3-9n^2\\[2ex]{\color{blue}{3\;\;\;\;\;\;2}}\;\\[2ex]\;{\color{red}{3}}\;\;\;\;\end{align*} Trinomial Degree of each term Degree of polynomial \begin{align*}x^2-7x+12 \\[2ex]{\color{blue}{2\;\;\;\;\;\;1\;\;\;\;\;\;0}}\;\\[2ex]\;\;\;\;\;\;{\color{red}{2}}\;\;\;\;\;\;\;\;\;\end{align*} \begin{align*}9a^2+6ab+b^2\\[2ex]{\color{blue}{2\;\;\;\;\;\;2\;\;\;\;\;\;2}}\;\;\\[2ex]\;\;\;\;\;\;{\color{red}{2}}\;\;\;\;\;\;\;\;\;\;\end{align*} \begin{align*}6m^4-m^3n^2+8mn^5\\[2ex]{\color{blue}{4\;\;\;\;\;\;\;\;\;\;5\;\;\;\;\;\;\;\;6}}\;\;\;\;\;\\[2ex]\;\;\;\;\;\;{\color{red}{6}}\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\end{align*} \begin{align*}z^4+3z^2-1\\[2ex]{\color{blue}{4\;\;\;\;\;\;2\;\;\;\;\;\;0}}\;\\[2ex]\;\;\;\;\;\;{\color{red}{4}}\;\;\;\;\;\;\;\;\;\end{align*} Polynomial Degree of each term Degree of polynomial \begin{align*}b+1\\[2ex]{\color{blue}{1\;\;\;\;0}}\\[2ex]\;{\color{red}{1}}\;\;\;\end{align*} \begin{align*}4y^2-7y+2\\[2ex]{\color{blue}{2\;\;\;\;\;\;1\;\;\;\;\;\;0}}\\[2ex]\;\;\;\;\;\;{\color{red}{2}}\;\;\;\;\;\;\;\;\end{align*} \begin{align*}4x^4+x^3+8x^2-9x+1\\[2ex]{\color{blue}{4\;\;\;\;\;\;3\;\;\;\;\;\;\;2\;\;\;\;\;\;\;\;1\;\;\;\;\;0}}\\[2ex]\;\;\;\;\;\;{\color{red}{4}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{align*}

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

Example 5.1.2

Find the degree of the following polynomials.

a. $10y$
b. $4{x}^{3}-7x+5$
c. $-15$
d. $-8{b}^{2}+9b-2$
e. $8x{y}^{2}+2y$

Solution

a. The exponent of $y$ is one.$y={y}^{1}$

$10y$
The degree is $1$.

b. The highest degree of all the terms is $3$.

$4{x}^{3}-7x+5$
The degree is $3$.

c. The degree of a constant is $0$.

$-15$
The degree is $0$.

d. The highest degree of all the terms is $2$.

$-8{b}^{2}+9b-2$
The degree is $2$.

e. The highest degree of all the terms is $3$.

$6m^4-m^3+8m$
The degree is $3$.

Try It

6) Find the degree of the following polynomials:

a. $-15b$
b. $10{z}^{4}+4{z}^{2}-5$
c. $12{c}^{5}{d}^{4}+9{c}^{3}{d}^{9}-7$
d. $3{x}^{2}y-4x$
e.$-9$

Solution

a. 1
b. 4
c. 12
d. 3
e. 0

7) Find the degree of the following polynomials:

a. $52$
b. ${a}^{4}b-17{a}^{4}$
c. $5x+6y+2z$
d. $3{x}^{2}-5x+7$
e. ${−}{a}^{3}$

Solution

a. 0
b. 5
c. 1
d. 2
e. 3

You 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 coefficient.

Example 5.1.3

Add: $25{y}^{2}+15{y}^{2}$.

Solution

Step 1: Combine like terms.

$40{y}^{2}$

Try It

8) Add: $12{q}^{2}+9{q}^{2}$.

Solution

$21{q}^{2}$

9) Add: $-15{c}^{2}+8{c}^{2}$.

Solution

$-7{c}^{2}$

Example 5.1.4

Subtract: $16p-(-7p)$

Solution

Step 1: Combine like terms.

$23p$

Try It

10) Subtract: $8m-(-5m)$

Solution

$13m$

11) Subtract: $-15{z}^{3}-(-5{z}^{3})$

Solution

$-10{z}^{3}$

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

Example 5.1.5

Simplify: ${c}^{2}+7{d}^{2}-6{c}^{2}$

Solution

Step 1: Combine like terms.

$-5{c}^{2}+7{d}^{2}$

Try It

12) Add: $8{y}^{2}+3{z}^{2}-3{y}^{2}$.

Solution

$5{y}^{2}+3{z}^{2}$

13) Add: $3{m}^{2}+{n}^{2}-7{m}^{2}$

Solution

$-4{m}^{2}+{n}^{2}$

Example 5.1.6

Simplify: ${u}^{2}v+5{u}^{2}-3{v}^{2}$

Solution

Step 1: There are no like terms to combine.

${u}^{2}v+5{u}^{2}-3{v}^{2}$

Try It

14) Simplify: ${m}^{2}{n}^{2}-8{m}^{2}+4{n}^{2}$

Solution

There are no like terms to combine.

15) Simplify: $p{q}^{2}-6p-5{q}^{2}$

Solution

There are no like terms to combine.

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 5.1.7

Find the sum: $(5{y}^{2}-3y+15)+(3{y}^{2}-4y-11)$

Solution

Step 1: Identify like terms.

$\left({\color{red}{5y^2}}\;-\;{\color{blue}{3y}}\;+\;{\color{purple}{15}}\right)\;+\;\left({\color{red}{3^2}}\;-\;{\color{blue}{4y}}\;-\;{\color{purple}{11}}\right)$

Step 2: Rearrange to get the like terms together.

${\color{blue}{5y^2+3y^2}}-{\color{red}{3y-4y}}+{\color{purple}{15-11}}$

Step 3: Combine like terms.

$8y^2-7y+4$

Try It

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

Solution

$8{x}^{2}-11x+1$

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

Solution

$17{y}^{2}+14y+1$

Example 5.1.8

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

Solution

Step 1: Distribute and identify like terms.

${\color{blue}{9w^2}}-{\color{red}{7w}}+{\color{purple}{5}}-{\color{blue}{2w^2}}+{\color{purple}{4}}$

Step 2: Rearrange the terms.

${\color{blue}{9w^2-2w^2}}-{\color{red}{7w}}+{\color{purple}{5}}+{\color{purple}{4}}$

Step 3: Combine like terms.

$7w^2-7w+9$

Try It

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

Solution

$15{x}^{2}+3x-5$

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

Solution

$6{b}^{2}+3$

Example 5.1.9

Subtract: $({c}^{2}-4c+7)$ from $(7{c}^{2}-5c+3)$.

Solution

Step 1: Write the equation.

$(7c^2-5c+3)-(c^2-4c+7)$

Step 2: Distribute and identify like terms.

${\color{blue}{7c^2}}-{\color{red}{5c}}+{\color{purple}{3}}-{\color{blue}{c^2}}+{\color{red}{4c}}-{\color{purple}{7}}$

Step 2: Rearrange the terms.

${\color{blue}{7c^2-c^2}}-{\color{red}{5c+4c}}+{\color{purple}{3+7}}$

Step 3: Combine like terms.

$6c^2-c-4$

Try It

20) Subtract: $(5{z}^{2}-6z-2)$ from $(7{z}^{2}+6z-4)$.

Solution

$2{z}^{2}+12z-2$

21) Subtract: $({x}^{2}-5x-8)$ from $(6{x}^{2}+9x-1)$.

Solution

$5{x}^{2}+14x+7$

Example 5.1.10

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

Solution

Step 1: Distribute.

${u}^{2}-6uv+5{v}^{2}+3{u}^{2}+2uv$

Step 2: Rearrange the terms, to put like terms together.

${u}^{2}+3{u}^{2}-6uv+2uv+5{v}^{2}$

Step 3: Combine like terms.

$4{u}^{2}-4uv+5{v}^{2}$

Try It

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

Solution

$5{x}^{2}-5xy+5{y}^{2}$

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

Solution

$7{x}^{2}-6xy-2{y}^{2}$

Example 5.1.11

Find the difference: $({p}^{2}+{q}^{2})-({p}^{2}+10pq-2{q}^{2})$.

Solution

Step 1: Distribute.

${p}^{2}+{q}^{2}-{p}^{2}-10pq+2{q}^{2}$

Step 2: Rearrange the terms, to put like terms together.

${p}^{2}-{p}^{2}-10pq+{q}^{2}+2{q}^{2}$

Step 3: Combine like terms.

$-10p{q}^{2}+3{q}^{2}$

Try It

24) Find the difference: $({a}^{2}+{b}^{2})-({a}^{2}+5ab-6{b}^{2})$.

Solution

$-5ab-5{b}^{2}$

25) Find the difference: $({m}^{2}+{n}^{2})-({m}^{2}-7mn-3{n}^{2})$.

Solution

$4{n}^{2}+7mn$

Example 5.1.12

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

Solution

Step 1: Distribute.

${a}^{3}-{a}^{2}b-a{b}^{2}-{b}^{3}+{a}^{2}b+a{b}^{2}$

Step 2: Rearrange the terms, to put like terms together.

${a}^{3}-{a}^{2}b+{a}^{2}b-a{b}^{2}+a{b}^{2}-{b}^{3}$

Step 3: Combine like terms.

${a}^{3}-{b}^{3}$

Try It

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

Solution

${x}^{3}-{y}^{3}$

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

Solution

${p}^{3}-2{p}^{2}q+{q}^{3}$

Evaluate a Polynomial for a Given Value

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 5.1.13

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

a. $x=4$
b. $x=-2$
c. $x=0$

Solution

a.
Step 1: Substitute 4 for $x$.

$5\left({\color{red}{4}}\right)^2-8\left({\color{red}{4}}\right)+4$

Step 2:  Simplify the exponents.

$5\cdot16-8\left(4\right)+4$

Step 3: Multiply.

$80-32+4$

Step 4: Simplify.

$52$

b.
Step 1: Substitute $-2$ for $x$.

$5\left({\color{red}{-2}}\right)^2-8\left({\color{red}{-2}}\right)+4$

Step 2: Simplify the exponents.

$5\cdot4-8(-2)+4$

Step 3: Multiply.

$20+16+4$

Step 4: Simplify.

40

c.
Step 1: Substitute $0$ for $x$.

$5\left({\color{red}{0}}\right)^2-8\left({\color{red}{0}}\right)+4$

Step 2: Simplify the exponents.

$5\cdot0-8(0)+4$

Step 3: Multiply.

$0+0+4$

Step 4: Simplify.

$4$

Try It

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

a. $x=3$
b. $x=-5$
c. $x=0$

Solution

a. $18$
b. $50$
c. $-15$

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

a. $z=-2$
b. $z=0$
c. $z=2$

Solution

a. $18$
b. $-4$
c. $14$

Example 5.1.14

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

Step 1: Substitute $t=2$.

$-16{(2)}^{2}+250$

Step 2: Simplify.

\begin{align*} &-16\times4+250\\ \text{Simplify.}\;\;&-64+250\\ &=186 \end{align*}

After $2$ seconds the height of the ball is $186$ feet.

Try It

30) 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.

Solution

$250$

31) 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.

Solution

$106$

Example 5.1.15

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

Step 1: Substitute $x=4$, $y=6$.

$6\left({\color{red}{4}}\right)^2+15\left({\color{red}{4}}\right){\color{blue}{\left(6\right)}}$

Step 2: Simplify.

\begin{align*}&6\cdot16+15\left({\color{red}{4}}\right){\color{blue}{\left(6\right)}}\\\text{Simplify.}\;\;&96+360\\&=456\end{align*}

The cost of producing the box is $456$.

Try It

32) 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.

Solution

$576 33) 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. Solution$750

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

Key Concepts

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

Self Check

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

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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 exponent of its variable.
monomial
A monomial is a term of the form $a{x}^{m}$, where $a$ is a constant and $m$ is a whole number; a monomial has exactly one term.
polynomial
A polynomial is a monomial, or two or more monomials combined by addition or subtraction.
standard form
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.
definition