3.1: Polynomials Review

Learning Objectives

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

• Determine the degree of polynomials
• Add and subtract polynomials
• Evaluate a polynomial function for a given value

Be Prepared

Before you get started, take this readiness quiz.

1. Simplify $3x^2+3x+1+8x^2+5x+5.$
2. Subtract $(5n+8)−(2n−1).$
3. Evaluate $4xy^2$ when $x=−2x$ and $y=5.$

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 $ax^m$, where $a$ is a constant, $x$ is the variable, and $m$ is a positive integer, it is called a monomial in one variable. Some examples of monomials in one variable are $2x^5$ and $-3x^{10}$. Monomials can also have more than one variable such as $−4a^2b^3c^2.$

Definition $\PageIndex{1}$

A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form $ax^m$, where $a$ is a constant and $m$ is a positive integer.

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.

Definition $\PageIndex{2}$

• A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.
• A polynomial with exactly one term is called a monomial.
• A polynomial with exactly two terms is called a binomial.
• A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

Polynomial	$y+1$	$4a^2−7ab+2b^2$	$4x^4+x^3+8x^2−9x+1$	$0$
Monomial	$14$	$8y^2$	$−9x^3y^5$	$−13a^3b^2c$
Binomial	$a+7ba+7b$	$4x^2−y^2$	$y^2−16$	$3p^3q−9p^2q$
Trinomial	$x^2−7x+12$	$9m^2+2mn−8n^2$	$6k^4−k^3+8k$	$z^4+3z^2−1$

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.

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.

Definition $\PageIndex{3}$

• 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 among 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 $8ab^2$ 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+2,1+2, is 3 so the degree is 3.

Here are some additional examples.

Monomial	$14$	$8ab^2$	$-9x^3y^5$	$-13a$
Degree	$0$	$3$	$8$	$1$

Binomial	$h+7$	$7b^2-3b$	$x^2y^2-25$	$4n^3-8n^2$
Degree of each term	$1,\;0$	$2,\; 1$	$4,\;0$	$3,\; 2$
Degree of the polynomial	$1$	$2$	$4$	$3$

Trinomial	$x^2-12x+27$	$9a^2+6ab+b^2$	$6m^4-m^3n^2+8mn^5$	$z^4+3z^2-1$
Degree of each term	$2,\; 1,\; 0$	$2,\; 2,\; 2$	$4,\; 5,\; 6$	$4,\; 2,\; 0$
Degree of the polynomial	$2$	$2$	$6$	$4$

Polynomial	$y-1$	$3y^2-2y-5$	$4x^4 +x^3 +8x^2-9x+1$
Degree of each term	$1,\; 0$	$2,\;1,\; 0$	$4, \;3, \;2, \;1,\; 0$
Degree of the polynomial	$1$	$2$	$4$

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 $\PageIndex{4}$

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

a. $7y^2−5y+3$

b. $−2a^4b^2$

c. $3x^5−4x^3−6x^2+x−8$

d. $2y−8xy^3$

e. $15$

Solution

	Polynomial	Number of terms	Type	Degree of terms	Degree of polynomial
a.	$7y^2−5y+3$	3	Trinomial	2, 1, 0	2
b.	$−2a^4b^2$	1	Monomial	4, 2	6
c.	$3x^5−4x^3−6x^2+x−8$	5	Polynomial	5, 3, 2, 1, 0	5
d.	$2y−8xy^3$	2	Binomial	1, 4	4
e.	$15$	1	Monomial	0	0

Try It $\PageIndex{5}$

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

a. $−5$

b. $8y^3−7y^2−y−3$

c. $−3x^2y−5xy+9xy^3$

d. $81m^2−4n^2$

e. $−3x^6y^3z$

Answer a

It is a monomial of degree 0.

Answer b

It is a polynomial of degree 3.

Answer c

It is a trinomial of degree 3.

Answer d

It is a binomial of degree 2.

Answer e

It is a monomial of degree 10.

Try It $\PageIndex{6}$

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

a. $64k^3−8$

b. $9m^3+4m^2−2$

c. $56$

d. $8a^4−7a^3b−6a^2b^2−4ab^3+7b^4$

e. $-p^4q^3$

Answer

a. It is a binomial of degree 3.

b. It is a trinomial of degree 3.

c. It is a monomial of degree 0.

d. It is a polynomial of degree 4.

e. It is a monomial of degree 7.

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 exponents. 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 $\PageIndex{7}$

Add or subtract:

a. $25y^2+15y^2$

b. $16pq^3−(−7pq^3)$

Solution

a.

	$\quad 25y^2+15y^2$
Combine like terms.	$=40y^2$

b.

	$\quad 16pq^3−(−7pq^3)$
Combine like terms.	$=23pq^3$)

Try It $\PageIndex{8}$

Add or subtract:

a. $12q^2+9q^2$

b. $8mn^3−(−5mn^3)$

Answer

a. $21q^2$

b. $13mn^3$

Try It $\PageIndex{9}$

Add or subtract:

a. $−15c^2+8c^2$

b. $−15y^2z^3−(−5y^2z^3)$

Answer

a. $−7c^2$

b. $−10y^2z^3$

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

Example $\PageIndex{10}$

Simplify:

a. $a^2+7b^2−6a^2$

b. $u^2v+5u^2−3v^2$

Solution

a.

	$\quad a^2+7b^2−6a^2$
Combine like terms.	$= −5a^2+7b^2$

b.

	$u^2v+5u^2−3v^2$
Combine like terms.	There are no like terms to combine. In this case, the polynomial is unchanged. $u^2v+5u^2−3v^2$

Try It $\PageIndex{11}$

Add:

a. $8y^2+3z^2−3y^2$

b. $m^2n^2−8m^2+4n^2$

Answer

a. $5y^2+3z^2$
b. $m^2n^2−8m^2+4n^2$

Try It $\PageIndex{12}$

Add:

a. $3m^2+n^2−7m^2$

b. $pq^2−6p−5q^2$

Answer

a. $−4m^2+n^2$
b. $pq^2−6p−5q^2$

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 $\PageIndex{13}$

Find the sum $(7y^2−2y+9)\;+\;(4y^2−8y−7)$.

Solution

	$\quad (7y^2−2y+9)\;+\;(4y^2−8y−7)$
Identify like terms.	$=(\underline{\underline{7y^2}}−\underline{2y}+9)+(\underline{\underline{4y^2}}−\underline{8y}−7)$
Rewrite without the parentheses, rearranging to get the like terms together.	$=\underline{\underline{7y^2+4y^2}}−\underline{2y−8y}+9−7$
Combine like terms.	$=11y^2−10y+2$

Try It $\PageIndex{14}$

Find the sum $(7x^2−4x+5)\;+\;(x^2−7x+3)$.

Answer

$8x^2−11x+8$

Try It $\PageIndex{15}$

Find the sum $(14y^2+6y−4)\;+\;(3y^2+8y+5)$.

Answer

$17y^2+14y+1$

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

Example $\PageIndex{16}$

Find the difference $(9w^2−7w+5)\;−\;(2w^2−4)$.

Solution

	$\quad (9w^2−7w+5)\;−\;(2w^2−4)$
Distribute and identify like terms.	$=\underline{\underline{9w^2}}−\underline{7w}+5-\underline{\underline{2w^2}}+4$
Rearrange the terms.	$= \underline{\underline{9w^2-2w^2}}−\underline{7w}+5+4$
Combine like terms.	$= 7w^2−7w+9$

Try It $\PageIndex{17}$

Find the difference $(8x^2+3x−19)\;−\;(7x^2−14)$.

Answer

$x^2+3x−5$

Try It $\PageIndex{18}$

Find the difference $(9b^2−5b−4)\;−\;(3b^2−5b−7)$.

Answer

$6b^2+3$

Example $\PageIndex{19}$

Subtract $p^2+10pq−2q^2$ from $p^2+q^2$.

Solution

	$\quad (p^2+q^2)\;−\;(p^2+10pq−2q^2)$
Distribute and identify like terms.	$=\underline{\underline{p^2}}+\underline{q^2}-\underline{\underline{p^2}}-10pq + \underline{2q^2}$
Rearrange the terms, putting like terms together.	$= \underline{\underline{p^2-p^2}}−10pq +\underline{q^2 + 2q^2}$
Combine like terms.	$=−10pq+3q^2$

Try It $\PageIndex{20}$

Subtract $a^2+5ab−6b^2$ from $a^2+b^2$.

Answer

$−5ab+7b^2$

Try It $\PageIndex{21}$

Subtract $m^2−7mn−3n^2$ from $m^2+n^2$.

Answer

$7mn+4n^2$

Example $\PageIndex{22}$

Find the sum $(u^2−6uv+5v^2)\;+\;(3u^2+2uv)$.

Solution

	$\quad (u^2−6uv+5v^2)\;+\;(3u^2+2uv)$
Distribute and identify like terms.	$=\underline{\underline{u^2}}-\underline{6uv}+5v^2+\underline{\underline{3u^2}}+ \underline{2uv}$
Rearrange the terms to put like terms together.	$=\underline{\underline{u^2}}+\underline{\underline{3u^2}}- \underline{6uv}+ \underline{2uv}+5v^2$
Combine like terms.	$=4u^2−4uv+5v^2$

Try It $\PageIndex{23}$

Find the sum $(3x^2−4xy+5y^2)\;+\;(2x^2−xy)$.

Answer

$5x^2−5xy+5y^2$

Try It $\PageIndex{24}$

Find the sum $(2x^2−3xy−2y^2)\;+\;(5x^2−3xy)$.

Answer

$7x^2−6xy−2y^2$

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

Example $\PageIndex{25}$

Simplify $(a^3−a^2b)\;−\;(ab^2+b^3)\;+\;(a^2b+ab^2)$.

Solution

	$\quad (a^3−a^2b)\;−\;(ab^2+b^3)\;+\;(a^2b+ab^2)$
Distribute.	$=a^3−a^2b − ab^2 - b^3 + a^2b+ab^2$
Rearrange the terms to put like terms together.	$=a^3−a^2b + a^2b− ab^2 + ab^2 - b^3$
Combine like terms.	$=a^3−b^3$

Try It $\PageIndex{26}$

Simplify $(x^3−x^2y)\;−\;(xy^2+y^3)\;+\;(x^2y+xy^2)$.

Answer

$x^3+y^3$

Try It $\PageIndex{27}$

Simplify $(p^3−p^2q)\;+\;(pq^2+q^3)\;−\;(p^2q+pq^2)$.

Answer

$p^3−3p^2q+q^3$

Evaluate a Polynomial

Example $\PageIndex{28}$

For the polynomial $5x^2−8x+4$ evaluate where:

a. $x=4$

b. $x=-2$

c. $x=0$

Solution

a.

	$\quad 5x^2−8x+4$
Substitute 4 for $x$	$\quad 5(4)^2−8(4)+4$
Simplify the exponents.	$=5\cdot 16−8(4)+4$
Multiply.	$=80-32+4$
Simplify.	$=52$

b.

	$\quad 5x^2−8x+4$
To find $f(-2)$, substitute $-2$ for $x$.	$\quad 5(-2)^2−8(-2)+4$
Simplify the exponents.	$= 5\cdot 4−8(-2)+4$
Multiply.	$= 20+16+4$
Simplify.	$=40$

c.

	$\quad 5x^2−8x+4$
To find $f(0)$, substitute $0$ for $x$.	$\quad 5(0)^2−8(0)+4$
Simplify the exponents.	$=5\cdot 0−8(0)+4$
Multiply.	$=0+0+4$
Simplify.	$=4$

Try It $\PageIndex{29}$

For the polynomial $3x^2+2x−15$, evaluate at

a. $x=3$

b. $x=-5$

c. $x=0$

Answer

a. $18$

b. $50$

c. $−15$

Try It $\PageIndex{30}$

For the polynomial $5x^2−x−4$, evaluate at

a. $x=-2$

b. $x=-1$

c. $x=0$

Answer

a. $20$

b. $2$

c. $−4$

Polynomials similar to the one in the next example are used in many fields to model the height of an object at some time after it is projected into the air. The polynomial in the next example is used specifically to model the height of an object which is dropped from 250 ft.

Example $\PageIndex{31}$

The polynomial $−16t^2+250$ 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

If we call the height $h$, then the solutions to the equation $h=−16t^2+250$ are $(t,h)$ where $h$ is the height in feet of the ball at time $t$ seconds.

	$−16t^2+250$
To find the height at $2$ seconds, we substitute $2$ for $t$.	$h=−16(2)^2+250$
Simplify.	$\quad=−16\cdot 4+250$
Simplify.	$\quad =−64+250$
Simplify.	$\quad =186$
Answer the question.	After 2 seconds the height of the ball is 186 feet. That is, from finding the solution $(t,h)=(2,186)$, we conclude that the height of the ball after 2 seconds is 186 feet.

Note that in the above example, the interpretation of the polynomial leads us to write the equation that relates height and time that we claim is true. So its solutions are of interest. We are asked, in particular, about the height after 2 seconds, so we proceed to find a solution where $t=2$. Our answer is the corresponding $h$-coordinate.

Try It $\PageIndex{32}$

The polynomial $−16t^2+150$ 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).

Answer

The height is $150$ feet.

Try It $\PageIndex{33}$

The polynomial $−16t^2+175$ 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.

Answer

The height is $31$ feet.

Key Concepts

• Monomial
  • A monomial is an algebraic expression with one term.
  • A monomial in one variable is a term of the form $ax^m$ where $a$ is a constant and $m$ is a whole number.
• 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.
• 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 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 axm,axm, 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.