6.1: 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

Quiz

Before you get started, take this readiness quiz.

1. Simplify: $8x+3x$.
   If you missed this problem, review Example 1.3.37.
2. Subtract: $(5n+8)−(2n−1)$.
   If you missed this problem, review Example 1.10.52.
3. Write in expanded form: $a^{5}$.
   If you missed this problem, review Example 1.3.7.

Identify Polynomials, Monomials, Binomials 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 $ax^{m}$, where $a$ is a constant and $m$ is a whole number, it is called a monomial. Some examples of monomial are $8,−2x^{2},4y^{3}$, and $11z^{7}$.

Definition: Monomials

A monomial is a term of the form $ax^{m}$, 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.

Definitions: 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.

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

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 $\PageIndex{1}$

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

1. $4y^{2}−8y−6$
2. $−5a^{4}b^{2}$
3. $2x^{5}−5x^{3}−3x + 4$
4. $13−5m^{3}$
5. q

Solution

$\begin{array}{lll}&{\text { Polynomial }} & {\text { Number of terms }} & {\text { Type }} \\ {\text { (a) }} & {4 y^{2}-8 y-6} & {3} & {\text { Trinomial }} \\ {\text { (b) }} & {-5 a^{4} b^{2}} & {1} & {\text { Monomial }} \\ {\text { (c) }} & {2 x^{5}-5 x^{3}-9 x^{2}+3 x+4} & {5} & {\text { Ponomial }} \\ {\text { (d) }} & {13-5 m^{3}} & {2} & {\text { Binomial }} \\ {\text { (e) }} & {q} & {1} & {\text { Monomial }}\end{array}$

Try It $\PageIndex{2}$

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

1. 5b
2. $8 y^{3}-7 y^{2}-y-3$
3. $-3 x^{2}-5 x+9$
4. $81-4 a^{2}$
5. $-5 x^{6}$

Answer

1. monomial
2. polynomial
3. trinomial
4. binomial
5. monomial

Try It $\PageIndex{3}$

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

1. $27 z^{3}-8$
2. $12 m^{3}-5 m^{2}-2 m$
3. $\frac{5}{6}$
4. $8 x^{4}-7 x^{2}-6 x-5$
5. $-n^{4}$

Answer

1. binomial
2. trinomial
3. monomial
4. polynomial
5. 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, i.e., it has no variable.

Definition: 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.

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 $\PageIndex{4}$

Find the degree of the following polynomials.

1. 10y
2. $4 x^{3}-7 x+5$
3. −15
4. $-8 b^{2}+9 b-2$
5. $8 x y^{2}+2 y$

Solution

1. $\begin{array}{ll} & 10y\\ \text{The exponent of y is one. } y=y^1 & \text{The degree is 1.}\end{array}$
2. $\begin{array}{ll} & 4 x^{3}-7 x+5\\ \text{The highest degree of all the terms is 3.} & \text{The degree is 3.}\end{array}$
3. $\begin{array}{ll} & -15\\ \text{The degree of a constant is 0.} & \text{The degree is 0.}\end{array}$
4. $\begin{array}{ll} & -8 b^{2}+9 b-2\\ \text{The highest degree of all the terms is 2.} & \text{The degree is 2.}\end{array}$
5. $\begin{array}{ll} & 8 x y^{2}+2 y\\ \text{The highest degree of all the terms is 3.} & \text{The degree is 3.}\end{array}$

Try It $\PageIndex{5}$

Find the degree of the following polynomials:

1. −15b
2. $10 z^{4}+4 z^{2}-5$
3. $12 c^{5} d^{4}+9 c^{3} d^{9}-7$
4. $3 x^{2} y-4 x$
5. −9

Answer

1. 1
2. 4
3. 12
4. 3
5. 0

Try It $\PageIndex{6}$

Find the degree of the following polynomials:

1. 52
2. $a^{4} b-17 a^{4}$
3. $5 x+6 y+2 z$
4. $3 x^{2}-5 x+7$
5. $-a^{3}$

Answer

1. 0
2. 5
3. 1
4. 2
5. 3

Add and Subtract Monomials

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 $\PageIndex{7}$

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

Solution

$\begin{array}{ll} & 25 y^{2}+15 y^{2}\\ \text{Combine like terms.} & 40y^{2}\end{array}$

Try It $\PageIndex{8}$

Add: $12 q^{2}+9 q^{2}$

Answer

21$q^{2}$

Try It $\PageIndex{9}$

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

Answer

$-7 c^{2}$

Example $\PageIndex{10}$

Subtract: 16p−(−7p)

Solution

$\begin{array}{ll} & 16p−(−7p) \\ \text{Combine like terms.} & 23p\end{array}$

Try It $\PageIndex{11}$

Subtract: 8m−(−5m).

Answer

13m

Try It $\PageIndex{12}$

Subtract: $-15 z^{3}-\left(-5 z^{3}\right)$

Answer

$-10 z^{3}$

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

Example $\PageIndex{13}$

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

Solution

$\begin{array}{ll} & c^{2}+7 d^{2}-6 c^{2} \\ \text{Combine like terms.} & -5 c^{2}+7 d^{2} \end{array}$

Try It $\PageIndex{14}$

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

Answer

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

Try It $\PageIndex{15}$

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

Answer

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

Example $\PageIndex{16}$

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

Solution

$\begin{array}{ll} &u^{2} v+5 u^{2}-3 v^{2} \\ \text{There are no like terms to combine.} & u^{2} v+5 u^{2}-3 v^{2} \end{array}$

Try It $\PageIndex{17}$

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

Answer

There are no like terms to combine.

Try It $\PageIndex{18}$

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

Answer

There are no like terms to combine.

Add and Subtract Polynomials

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 $\PageIndex{19}$

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

Solution

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

Try It $\PageIndex{20}$

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

Answer

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

Try It $\PageIndex{21}$

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

Answer

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

Example $\PageIndex{22}$

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

Solution

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

Try It $\PageIndex{23}$

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

Answer

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

Try It $\PageIndex{24}$

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

Answer

$6 b^{2}+3$

Example $\PageIndex{25}$

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

Solution

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

Try It $\PageIndex{26}$

Subtract: $\left(5 z^{2}-6 z-2\right)$ from $\left(7 z^{2}+6 z-4\right)$

Answer

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

Try It $\PageIndex{27}$

Subtract: $\left(x^{2}-5 x-8\right)$ from $\left(6 x^{2}+9 x-1\right)$

Answer

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

Example $\PageIndex{28}$

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

Solution

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

Try It $\PageIndex{29}$

Find the sum: $\left(3 x^{2}-4 x y+5 y^{2}\right)+\left(2 x^{2}-x y\right)$

Answer

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

Try It $\PageIndex{30}$

Find the sum: $\left(2 x^{2}-3 x y-2 y^{2}\right)+\left(5 x^{2}-3 x y\right)$

Answer

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

Example $\PageIndex{31}$

Find the difference: $\left(p^{2}+q^{2}\right)-\left(p^{2}+10 p q-2 q^{2}\right)$

Solution

$\begin{array}{ll} & {\left(p^{2}+q^{2}\right)-\left(p^{2}+10 p q-2 q^{2}\right)} \\ \text{Distribute.} &{p^{2}+q^{2}-p^{2}-10 p q+2 q^{2}} \\\text{Rearrange the terms, to put like terms together} & {p^{2}-p^{2}-10 p q+q^{2}+2 q^{2}} \\\text{Combine like terms.} & {-10 p q+3 q^{2}}\end{array}$

Try It $\PageIndex{32}$

Find the difference: $\left(a^{2}+b^{2}\right)-\left(a^{2}+5 a b-6 b^{2}\right)$

Answer

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

Try It $\PageIndex{33}$

Find the difference: $\left(m^{2}+n^{2}\right)-\left(m^{2}-7 m n-3 n^{2}\right)$

Answer

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

Example $\PageIndex{34}$

Simplify: $\left(a^{3}-a^{2} b\right)-\left(a b^{2}+b^{3}\right)+\left(a^{2} b+a b^{2}\right)$

Solution

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

Try It $\PageIndex{35}$

Simplify: $\left(x^{3}-x^{2} y\right)-\left(x y^{2}+y^{3}\right)+\left(x^{2} y+x y^{2}\right)$

Answer

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

Try It $\PageIndex{36}$

Simplify: $\left(p^{3}-p^{2} q\right)+\left(p q^{2}+q^{3}\right)-\left(p^{2} q+p q^{2}\right)$

Answer

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

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 $\PageIndex{37}$

Evaluate $5x^{2}−8x+4$ when

1. x=4
2. x=−2
3. x=0

Solution

1. x=4
Simplify the exponents.
Multiply.
Simplify.

2. x=−2
Simplify the exponents.
Multiply.
Simplify.

3. x=0
Simplify the exponents.
Multiply.
Simplify.

Try It $\PageIndex{38}$

Evaluate: $3x^{2}+2x−15$ when

1. x=3
2. x=−5
3. x=0

Answer

1. 18
2. 50
3. −15

Try It $\PageIndex{39}$

Evaluate: $5z^{2}−z−4$ when

1. z=−2
2. z=0
3. z=2

Answer

1. 18
2. −4
3. 14

Example $\PageIndex{40}$

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

Solution

$\begin{array}{ll } & −16t^{2}+250 \\ \text{Substitute t = 2.} & -16(2)^{2} + 250 \\ \text{Simplify }& −16\cdot 4+250 \\ \text{Simplify }& -64 + 250\\ \text{Simplify }& 186 \\& \text{After 2 seconds the height of the ball is 186 feet. } \end{array}$

Try It $\PageIndex{41}$

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

Answer

250

Try It $\PageIndex{42}$

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

Answer

106

Example $\PageIndex{43}$

The polynomial $6x^{2}+15xy$ 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=6y=6 feet.

Solution

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

Try It $\PageIndex{43}$

The polynomial $6x^{2}+15xy$ 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.

Answer

$576

Try It $\PageIndex{44}$

The polynomial $6x^{2}+15xy$ 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.

Answer

$750

Key Concepts

• Monomials
  • A monomial is a term of the form $ax^{m}$, where aa is a constant and mm 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.

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 $ax^m$, 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.