Add and Subtract Polynomials

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; and evaluate a polynomial for a given value.

Identify polynomials, monomials, binomials, and trinomials

You learned earlier that a term is a constant, or the product of a constant and one or more variables. The constant is called a coefficient. When a term is of the form $ax^m$ , where $a$ is a constant and $m$ is a whole number, it is called a monomial. A monomial, or a sum and/or difference of monomials, is called a polynomial.

Polynomials.

polynomial — a monomial, or two or more monomials, combined by addition or subtraction
monomial — a polynomial with exactly one term
binomial — a polynomial with exactly two terms
trinomial — a polynomial with exactly three terms

Notice the roots of these words: poly- means many, mono- means one, bi- means two, and tri- means three. Here are some examples:

	Example 1	Example 2	Example 3
Polynomial	$b + 1$	$4y^2 - 7y + 2$	$5x^5 - 4x^4 + x^3 + 8x^2 - 9x + 1$
Monomial	$5$	$4b^2$	$-9x^3$
Binomial	$3a - 7$	$y^2 - 9$	$17x^3 + 14x^2$
Trinomial	$x^2 - 5x + 6$	$4y^2 - 7y + 2$	$5a^4 - 3a^3 + a$

Notice that every monomial, binomial, and trinomial is also a polynomial — they are special members of the polynomial family, 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. Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial: (a) $8x^2 - 7x - 9$ (b) $-5a^4$ (c) $x^4 - 7x^3 - 6x^2 + 5x + 2$ (d) $11 - 4y^3$ (e) $n$

	Polynomial	Number of terms	Type
(a)	$8x^2 - 7x - 9$	3	Trinomial
(b)	$-5a^4$	1	Monomial
(c)	$x^4 - 7x^3 - 6x^2 + 5x + 2$	5	Polynomial
(d)	$11 - 4y^3$	2	Binomial
(e)	$n$	1	Monomial

Classify $2x^3 - 4x^2 - x - 8$ as a monomial, binomial, trinomial, or other polynomial. Enter the number of terms it has.

Classify $9x^3 - 5x^2 - x$ as a monomial, binomial, trinomial, or other polynomial. Enter the number of terms it has.

Determine the degree of polynomials

In this section, we will work with polynomials that have only one variable in each term. The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

Degree of a polynomial.

The degree of a term is the exponent of its variable.
The degree of a constant is $0$ .
The degree of a polynomial is the highest degree of all its terms.

A monomial that has no variable, just a constant, is a special case — the degree of a constant is $0$ because it has no variable. Remember: any base written without an exponent has an implied exponent of $1$ .

Let’s see how this works, starting with monomials and progressing to polynomials with more terms.

	Example 1	Example 2	Example 3	Example 4
Monomial	$5$	$4b^2$	$-9x^3$	$-18$
Degree	$0$	$2$	$3$	$0$
Binomial	$b+1$	$3a-7$	$y^2-9$	$17x^3+14x^2$
Degree	$1$	$1$	$2$	$3$
Trinomial	$x^2-5x+6$	$4y^2-7y+2$	$5a^4-3a^3+a$	$x^4+2x^2-5$
Degree	$2$	$2$	$4$	$4$

Example. Find the degree of the following polynomials: (a) $4x$ (b) $3x^3 - 5x + 7$ (c) $-11$ (d) $-6x^2 + 9x - 3$ (e) $8x + 2$

(a) The exponent of $x$ is one ( $x = x^1$ ), so the degree is $1$ . (b) The highest degree of all the terms is $3$ , so the degree is $3$ . (c) The degree of a constant is $0$ , so the degree is $0$ . (d) The highest degree of all the terms is $2$ , so the degree is $2$ . (e) The highest degree of all the terms is $1$ , so the degree is $1$ .

Find the degree of the polynomial $3x^4 + 4x^2 - 8$ .

Find the degree of the polynomial $y^5 - 5y^3 + y$ .

Working with polynomials is easier when you list the terms in descending order of degree. When a polynomial is written this way, it is said to be in standard form. Get in the habit of writing the term with the highest degree first.

Add and subtract monomials

You already know how to simplify expressions by combining like terms. Adding and subtracting monomials is the same as combining like terms — like terms must have the same variable raised to the same exponent. Recall that when we combine like terms, only the coefficients are combined, never the exponents.

Example. Add: $17x^2 + 6x^2$ .

17x^2 + 6x^2

Combine like terms:

23x^2

Add: $12x^2 + 5x^2$ .

Example. Subtract: $11n - (-8n)$ .

11n - (-8n)

Combine like terms:

19n

Subtract: $9n - (-5n)$ .

Example. Simplify: $a^2 + 4b^2 - 7a^2$ .

a^2 + 4b^2 - 7a^2

Combine like terms:

-6a^2 + 4b^2

Remember, $-6a^2$ and $4b^2$ are not like terms — the variables are not the same, so this expression cannot be simplified any further.

Add: $3x^2 + 3y^2 - 5x^2$ .

Add and subtract polynomials

Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms — those with the same variables raised to the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together. It may also be helpful to underline, circle, or box like terms.

Example. Find the sum: $(4x^2 - 5x + 1) + (3x^2 - 8x - 9)$ .

Identify like terms:

4x^2 - 5x + 1 + 3x^2 - 8x - 9

Rearrange to get the like terms together:

4x^2 + 3x^2 - 5x - 8x + 1 - 9

Combine like terms:

7x^2 - 13x - 8

Find the sum: $(3x^2 - 2x + 8) + (x^2 - 6x + 2)$ .

Find the sum: $(7y^2 + 4y - 6) + (4y^2 + 5y + 1)$ .

Parentheses are grouping symbols. When we add polynomials, we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.

Example. Find the difference: $(7u^2 - 5u + 3) - (4u^2 - 2)$ .

Distribute and identify like terms:

7u^2 - 5u + 3 - 4u^2 + 2

Rearrange the terms:

7u^2 - 4u^2 - 5u + 3 + 2

Combine like terms:

3u^2 - 5u + 5

Find the difference: $(6y^2 + 3y - 1) - (3y^2 - 4)$ .

Find the difference: $(8u^2 - 7u - 2) - (5u^2 - 6u - 4)$ .

Example. Subtract $(m^2 - 3m + 8)$ from $(9m^2 - 7m + 4)$ .

Distribute and identify like terms:

9m^2 - 7m + 4 - m^2 + 3m - 8

Rearrange the terms:

9m^2 - m^2 - 7m + 3m + 4 - 8

Combine like terms:

8m^2 - 4m - 4

Subtract $(4n^2 - 7n - 3)$ from $(8n^2 + 5n - 3)$ .

Subtract $(a^2 - 4a - 9)$ from $(6a^2 + 4a - 1)$ .

Evaluate a polynomial for a given value

Since polynomials are expressions, we follow the same procedures to evaluate a polynomial as we do for any expression: substitute the given value for the variable into the polynomial, and then simplify.

Example. Evaluate $3x^2 - 9x + 7$ when (a) $x = 3$ (b) $x = -1$ .

(a) Substitute $3$ for $x$ :

3(3)^2 - 9(3) + 7

Simplify the expression with the exponent:

3 \cdot 9 - 9(3) + 7

Multiply:

27 - 27 + 7

Simplify:

7

(b) Substitute $-1$ for $x$ :

3(-1)^2 - 9(-1) + 7

Simplify the expression with the exponent:

3 \cdot 1 - 9(-1) + 7

Multiply:

3 + 9 + 7

Simplify:

19

Evaluate $2x^2 + 4x - 3$ when $x = 2$ .

Evaluate $2x^2 + 4x - 3$ when $x = -3$ .

Example. The polynomial $-16t^2 + 300$ gives the height, in feet, of an object $t$ seconds after it is dropped from a $300$ -foot-tall bridge. Find the height after $t = 3$ seconds.

Substitute $3$ for $t$ :

-16(3)^2 + 300

Simplify the expression with the exponent:

-16 \cdot 9 + 300

Multiply:

-144 + 300

Simplify:

156

The height of the object is $156$ feet after $t = 3$ seconds.

The polynomial $-8t^2 + 24t + 4$ gives the height, in feet, of a ball $t$ seconds after it is tossed into the air from an initial height of 4 feet. Find the height after $t = 3$ seconds.

The polynomial $-8t^2 + 24t + 4$ gives the height, in feet, of a ball $t$ seconds after it is tossed into the air from an initial height of 4 feet. Find the height after $t = 2$ seconds.

Key terms

polynomial — a monomial, or two or more monomials, combined by addition or subtraction. monomial — a polynomial with exactly one term. binomial — a polynomial with exactly two terms. trinomial — a polynomial with exactly three terms. degree of a term — the exponent of its variable (the degree of a constant is $0$ ). degree of a polynomial — the highest degree of all its terms. standard form — a polynomial written with its terms in descending order of degree.

This section is adapted from Prealgebra 2e, Section 10.1: Add and Subtract Polynomials by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the polynomial-classification and degree tables as markdown tables; omitted the Be Prepared quiz, Media callout, Self Check checklist, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

Polynomials Use Multiplication Properties of Exponents