Before you get started, take this readiness quiz.
Simplify: $8x+3x.$
If you missed this problem, review Example 1.24.
Subtract: $\left(5n+8\right)-\left(2n-1\right).$
If you missed this problem, review Example 1.139.
Write in expanded form: ${a}^{5}.$
If you missed this problem, review Example 1.14.
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 $a{x}^{m}$, where $a$ is a constant and $m$ is a whole number, it is called a monomial. Some examples of monomial are $8,\mathrm{-2}{x}^{2},4{y}^{3},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}11{z}^{7}$.
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.
$\begin{array}{ccccccccccccc}\text{Polynomial}\hfill & & & \phantom{\rule{2em}{0ex}}b+1\hfill & & & \phantom{\rule{2em}{0ex}}4{y}^{2}-7y+2\hfill & & & \phantom{\rule{2em}{0ex}}4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1\hfill & & & \\ \text{Monomial}\hfill & & & \phantom{\rule{2em}{0ex}}14\hfill & & & \phantom{\rule{2em}{0ex}}8{y}^{2}\hfill & & & \phantom{\rule{2em}{0ex}}\mathrm{-9}{x}^{3}{y}^{5}\hfill & & & \phantom{\rule{2em}{0ex}}\mathrm{-13}\hfill \\ \text{Binomial}\hfill & & & \phantom{\rule{2em}{0ex}}a+7\hfill & & & \phantom{\rule{2em}{0ex}}4b-5\hfill & & & \phantom{\rule{2em}{0ex}}{y}^{2}-16\hfill & & & \phantom{\rule{2em}{0ex}}3{x}^{3}-9{x}^{2}\hfill \\ \text{Trinomial}\hfill & & & \phantom{\rule{2em}{0ex}}{x}^{2}-7x+12\hfill & & & \phantom{\rule{2em}{0ex}}9{y}^{2}+2y-8\hfill & & & \phantom{\rule{2em}{0ex}}6{m}^{4}-{m}^{3}+8m\hfill & & & \phantom{\rule{2em}{0ex}}{z}^{4}+3{z}^{2}-1\hfill \end{array}$
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 6.1
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.
- ⓐ $4{y}^{2}-8y-6$
- ⓑ $\mathrm{-5}{a}^{4}{b}^{2}$
- ⓒ $2{x}^{5}-5{x}^{3}-9{x}^{2}+3x+4$
- ⓓ $13-5{m}^{3}$
- ⓔ $q$
Polynomial | Number of terms | Type | |
ⓐ | $4{y}^{2}-8y-6$ | $3$ | Trinomial |
ⓑ | $\mathrm{-5}{a}^{4}{b}^{2}$ | $1$ | Monomial |
ⓒ | $2{x}^{5}-5{x}^{3}-9{x}^{2}+3x+4$ | $5$ | Polynomial |
ⓓ | $13-5{m}^{3}$ | $2$ | Binomial |
ⓔ | $q$ | $1$ | Monomial |
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:
ⓐ $5b$ ⓑ $8{y}^{3}-7{y}^{2}-y-3$ ⓒ $\mathrm{-3}{x}^{2}-5x+9$ ⓓ $81-4{a}^{2}$ ⓔ $\mathrm{-5}{x}^{6}$
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:
ⓐ $27{z}^{3}-8$ ⓑ $12{m}^{3}-5{m}^{2}-2m$ ⓒ $\frac{5}{6}$ ⓓ $8{x}^{4}-7{x}^{2}-6x-5$ ⓔ $\text{\u2212}{n}^{4}$
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.
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 6.2
Find the degree of the following polynomials.
- ⓐ $10y$
- ⓑ $4{x}^{3}-7x+5$
- ⓒ $\mathrm{-15}$
- ⓓ $\mathrm{-8}{b}^{2}+9b-2$
- ⓔ $8x{y}^{2}+2y$
ⓐ
The exponent of $y$ is one. $y={y}^{1}$ |
$10y$
The degree is 1. |
ⓑ
The highest degree of all the terms is 3. |
$4{x}^{3}-7x+5$
The degree is 3. |
ⓒ
The degree of a constant is 0. |
$\mathrm{-15}$
The degree is 0. |
ⓓ
The highest degree of all the terms is 2. |
$\mathrm{-8}{b}^{2}+9b-2$
The degree is 2. |
ⓔ
The highest degree of all the terms is 3. |
$8x{y}^{2}+2y$
The degree is 3. |
Find the degree of the following polynomials:
ⓐ $\mathrm{-15}b$ ⓑ $10{z}^{4}+4{z}^{2}-5$ ⓒ $12{c}^{5}{d}^{4}+9{c}^{3}{d}^{9}-7$ ⓓ $3{x}^{2}y-4x$ ⓔ $\mathrm{-9}$
Find the degree of the following polynomials:
ⓐ $52$ ⓑ ${a}^{4}b-17{a}^{4}$ ⓒ $5x+6y+2z$ ⓓ $3{x}^{2}-5x+7$ ⓔ $\text{\u2212}{a}^{3}$
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 6.3
Add: $25{y}^{2}+15{y}^{2}$.
$25{y}^{2}+15{y}^{2}$ | |
Combine like terms. | $40{y}^{2}$ |
Add: $12{q}^{2}+9{q}^{2}.$
Add: $\mathrm{-15}{c}^{2}+8{c}^{2}.$
Example 6.4
Subtract: $16p-\left(\mathrm{-7}p\right)$.
$16p-\left(\mathrm{-7}p\right)$ | |
Combine like terms. | $23p$ |
Subtract: $8m-\left(\mathrm{-5}m\right).$
Subtract: $\mathrm{-15}{z}^{3}-\left(\mathrm{-5}{z}^{3}\right).$
Remember that like terms must have the same variables with the same exponents.
Example 6.5
Simplify: ${c}^{2}+7{d}^{2}-6{c}^{2}$.
${c}^{2}+7{d}^{2}-6{c}^{2}$ | |
Combine like terms. | $\mathrm{-5}{c}^{2}+7{d}^{2}$ |
Add: $8{y}^{2}+3{z}^{2}-3{y}^{2}.$
Add: $3{m}^{2}+{n}^{2}-7{m}^{2}.$
Example 6.6
Simplify: ${u}^{2}v+5{u}^{2}-3{v}^{2}$.
${u}^{2}v+5{u}^{2}-3{v}^{2}$ | |
There are no like terms to combine. | ${u}^{2}v+5{u}^{2}-3{v}^{2}$ |
Simplify: ${m}^{2}{n}^{2}-8{m}^{2}+4{n}^{2}.$
Simplify: $p{q}^{2}-6p-5{q}^{2}.$
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 6.7
Find the sum: $\left(5{y}^{2}-3y+15\right)+\left(3{y}^{2}-4y-11\right).$
Identify like terms. | |
Rearrange to get the like terms together. | |
Combine like terms. |
Find the sum: $\left(7{x}^{2}-4x+5\right)+\left({x}^{2}-7x+3\right).$
Find the sum: $\left(14{y}^{2}+6y-4\right)+\left(3{y}^{2}+8y+5\right).$
Example 6.8
Find the difference: $\left(9{w}^{2}-7w+5\right)-\left(2{w}^{2}-4\right).$
Distribute and identify like terms. | |
Rearrange the terms. | |
Combine like terms. |
Find the difference: $\left(8{x}^{2}+3x-19\right)-\left(7{x}^{2}-14\right).$
Find the difference: $\left(9{b}^{2}-5b-4\right)-\left(3{b}^{2}-5b-7\right).$
Example 6.9
Subtract: $\left({c}^{2}-4c+7\right)$ from $\left(7{c}^{2}-5c+3\right)$.
Distribute and identify like terms. | |
Rearrange the terms. | |
Combine like terms. |
Subtract: $\left(5{z}^{2}-6z-2\right)$ from $\left(7{z}^{2}+6z-4\right)$.
Subtract: $\left({x}^{2}-5x-8\right)$ from $\left(6{x}^{2}+9x-1\right)$.
Example 6.10
Find the sum: $\left({u}^{2}-6uv+5{v}^{2}\right)+\left(3{u}^{2}+2uv\right)$.
$\left({u}^{2}-6uv+5{v}^{2}\right)+\left(3{u}^{2}+2uv\right)$ | |
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. | $4{u}^{2}-4uv+5{v}^{2}$ |
Find the sum: $\left(3{x}^{2}-4xy+5{y}^{2}\right)+\left(2{x}^{2}-xy\right)$.
Find the sum: $\left(2{x}^{2}-3xy-2{y}^{2}\right)+\left(5{x}^{2}-3xy\right)$.
Example 6.11
Find the difference: $\left({p}^{2}+{q}^{2}\right)-\left({p}^{2}+10pq-2{q}^{2}\right)$.
$\left({p}^{2}+{q}^{2}\right)-\left({p}^{2}+10pq-2{q}^{2}\right)$ | |
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. | $\mathrm{-10}pq+3{q}^{2}$ |
Find the difference: $\left({a}^{2}+{b}^{2}\right)-\left({a}^{2}+5ab-6{b}^{2}\right)$.
Find the difference: $\left({m}^{2}+{n}^{2}\right)-\left({m}^{2}-7mn-3{n}^{2}\right)$.
Example 6.12
Simplify: $\left({a}^{3}-{a}^{2}b\right)-\left(a{b}^{2}+{b}^{3}\right)+\left({a}^{2}b+a{b}^{2}\right)$.
$\left({a}^{3}-{a}^{2}b\right)-\left(a{b}^{2}+{b}^{3}\right)+\left({a}^{2}b+a{b}^{2}\right)$ | |
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. | ${a}^{3}-{b}^{3}$ |
Simplify: $\left({x}^{3}-{x}^{2}y\right)-\left(x{y}^{2}+{y}^{3}\right)+\left({x}^{2}y+x{y}^{2}\right)$.
Simplify: $\left({p}^{3}-{p}^{2}q\right)+\left(p{q}^{2}+{q}^{3}\right)-\left({p}^{2}q+p{q}^{2}\right)$.
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 6.13
Evaluate $5{x}^{2}-8x+4$ when
- ⓐ $x=4$
- ⓑ $x=\mathrm{-2}$
- ⓒ $x=0$
ⓐ $x=4$ | |
Simplify the exponents. | |
Multiply. | |
Simplify. |
ⓑ $x=\mathrm{-2}$ | |
Simplify the exponents. | |
Multiply. | |
Simplify. |
ⓒ $x=0$ | |
Simplify the exponents. | |
Multiply. | |
Simplify. |
Evaluate: $3{x}^{2}+2x-15$ when
- ⓐ $x=3$
- ⓑ $x=\mathrm{-5}$
- ⓒ $x=0$
Evaluate: $5{z}^{2}-z-4$ when
- ⓐ $z=\mathrm{-2}$
- ⓑ $z=0$
- ⓒ $z=2$
Example 6.14
The polynomial $\mathrm{-16}{t}^{2}+250$ 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.
$\mathrm{-16}{t}^{2}+250$ | |
Substitute $t=2$. | $\mathrm{-16}{(2)}^{2}+250$ |
Simplify. | $\mathrm{-16}\xb74+250$ |
Simplify. | $\mathrm{-64}+250$ |
Simplify. | $186$ |
After 2 seconds the height of the ball is 186 feet. |
The polynomial $\mathrm{-16}{t}^{2}+250$ 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.
The polynomial $\mathrm{-16}{t}^{2}+250$ 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.
Example 6.15
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.
Simplify. | |
Simplify. | |
Simplify. | |
The cost of producing the box is $456. |
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.
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.
Media Access Additional Online Resources
Access these online resources for additional instruction and practice with adding and subtracting polynomials.
Section 6.1 Exercises
Practice Makes Perfect
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.
ⓐ $81{b}^{5}-24{b}^{3}+1$
ⓑ $5{c}^{3}+11{c}^{2}-c-8$
ⓒ $\frac{14}{15}y+\frac{1}{7}$
ⓓ 5
ⓔ $4y+17$
ⓐ ${x}^{2}-{y}^{2}$ ⓑ $\mathrm{-13}{c}^{4}$ ⓒ ${x}^{2}+5x-7$ ⓓ ${x}^{2}{y}^{2}-2xy+8$ ⓔ 19
ⓐ $11{y}^{2}$ ⓑ $\mathrm{-73}$ ⓒ $6{x}^{2}-3xy+4x-2y+{y}^{2}$ ⓓ $4y+17$ ⓔ $5{c}^{3}+11{c}^{2}-c-8$
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
ⓐ $9{y}^{3}-10{y}^{2}+2y-6$ ⓑ $\mathrm{-12}{p}^{4}$ ⓒ ${a}^{2}+9a+18$ ⓓ $20{x}^{2}{y}^{2}-10{a}^{2}{b}^{2}+30$ ⓔ 17
ⓐ $62{y}^{2}$ ⓑ 15 ⓒ $6{x}^{2}-3xy+4x-2y+{y}^{2}$ ⓓ $10-9x$ ⓔ ${m}^{4}+4{m}^{3}+6{m}^{2}+4m+1$
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
${\phantom{\rule{0.2em}{0ex}}\text{4y}}^{3}+6{y}^{3}$
$\mathrm{-3}m+9m$
$\text{\u2212}y-5y$
$13z-(\mathrm{-4}z)$
$\mathrm{-10}x-35x$
$\text{14x}-3y-13x$
$5{u}^{2}+4{v}^{2}-6{u}^{2}$
$p{q}^{2}-4p-3{q}^{2}$
${x}^{2}y-3x+7x{y}^{2}$
$\text{19y}+5z$
Add: $\phantom{\rule{0.2em}{0ex}}\text{4x},3y,\mathrm{-3}x$
Subtract $2{p}^{4}\text{from}-7{p}^{4}$.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
$(4{y}^{2}+10y+3)+(8{y}^{2}-6y+5)$
$({y}^{2}+9y+4)+(\mathrm{-2}{y}^{2}-5y-1)$
$(7{x}^{2}-9x+2)+(6{x}^{2}-4)$
$({p}^{2}-6p-18)+(2{p}^{2}+11)$
$(3{b}^{2}-4b+1)-(5{b}^{2}-b-2)$
$({b}^{2}-7b+5)-({b}^{2}-2b+9)$
$(10{r}^{2}-20r)-(r-8)$
Subtract $\left(5{y}^{2}-y+12\right)$ from $\left(10{y}^{2}-8y-20\right)$.
Subtract $\left(5{x}^{2}-x+12\right)$ from $\left(9{x}^{2}-6x-20\right)$.
Find the sum of
$\left({q}^{2}+4q+13\right)$ and $\left(7{q}^{3}-3\right)$.
Find the sum of
$\left({b}^{2}+5b+13\right)$ and $\left(4{b}^{3}-6\right)$.
Find the difference of
$\left({z}^{2}-3z-18\right)$ and
$\left({z}^{2}+5z-20\right)$.
Find the difference of
$\left({t}^{2}-5t-15\right)$ and
$\left({t}^{2}+4t-17\right)$.
$\left(\mathrm{-5}{x}^{2}-4xy-3{y}^{2}\right)+\left(2{x}^{2}-7xy\right)$
$\left(2{r}^{2}-3rs-2{s}^{2}\right)+\left(5{r}^{2}-3rs\right)$
$\left({m}^{2}+2{n}^{2}\right)-\left({m}^{2}-8mn-{n}^{2}\right)$
$\left({j}^{2}-{k}^{2}\right)-\left({j}^{2}-8jk-5{k}^{2}\right)$
$\left({p}^{3}-3{p}^{2}q\right)+\left(2p{q}^{2}+4{q}^{3}\right)-\left(3{p}^{2}q+p{q}^{2}\right)$
$\left({a}^{3}-2{a}^{2}b\right)+\left(a{b}^{2}+{b}^{3}\right)-\left(3{a}^{2}b+4a{b}^{2}\right)$
$\left({x}^{3}-2{x}^{2}y\right)-\left(x{y}^{2}-3{y}^{3}\right)-\left({x}^{2}y-4x{y}^{2}\right)$
Evaluate a Polynomial for a Given Value
In the following exercises, evaluate each polynomial for the given value.
Evaluate $5{y}^{2}-y-7$ when:
ⓐ $y=\mathrm{-4}$ ⓑ $y=1$ ⓒ $y=0$
Evaluate $16-36{x}^{2}$ when:
ⓐ $x=\mathrm{-1}$ ⓑ $x=0$ ⓒ $x=2$
A painter drops a brush from a platform 75 feet high. The polynomial $\mathrm{-16}{t}^{2}+75$ gives the height of the brush $t$ seconds after it was dropped. Find the height after $t=2$ seconds.
A girl drops a ball off a cliff into the ocean. The polynomial $\mathrm{-16}{t}^{2}+250$ 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.
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 $\mathrm{-4}{p}^{2}+420p.$ Find the revenue received when $p=60$ dollars.
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 $\mathrm{-4}{p}^{2}+420p.$ Find the revenue received when $p=90$ dollars.
Everyday Math
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 $-\frac{1}{150}{x}^{2}+\frac{1}{3}x$. Find the fuel efficiency when $x=30\phantom{\rule{0.2em}{0ex}}\text{mph}$.
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 $x=40\phantom{\rule{0.2em}{0ex}}\text{mph}$.
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.
Height of Projectile The height (in feet) of an object projected upward is given by the polynomial $\mathrm{-16}{t}^{2}+60t+90$ where $t$ represents time in seconds. Find the height after $t=2.5$ seconds.
Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial $\frac{9}{5}c+32$ where $c$ represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when $c=65\text{\xb0}.$
Writing Exercises
Using your own words, explain the difference between a monomial, a binomial, and a trinomial.
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?
Jonathan thinks that $\frac{1}{3}$ and $\frac{1}{x}$ are both monomials. What is wrong with his reasoning?
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. 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.