7.3: Adding and Subtracting Polynomials

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    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

    Be Prepared 10.1

    Before you get started, take this readiness quiz.

    Simplify:
    If you missed this problem, review Example 2.22.

    Be Prepared 10.2

    Subtract:
    If you missed this problem, review Example 7.29.

    Be Prepared 10.3

    Evaluate: when
    If you missed this problem, review Example 2.18.

    Identify Polynomials, Monomials, Binomials, and Trinomials

    In Evaluate, Simplify, and Translate Expressions, you learned that a term is a constant or the product of a constant and one or more variables. When it is of the form where is a constant and 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:

    • poly- means many
    • mono- means one
    • bi- means two
    • tri- means three

    Here are some examples of polynomials:

    Polynomial
    Monomial
    Binomial
    Trinomial

    Notice that every monomial, binomial, and trinomial is also a polynomial. They are 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 10.1

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

    1. ⓐ
    2. ⓑ
    3. ⓒ
    4. ⓓ
    5. ⓔ
    Answer

    Polynomial Number of terms Type
    ⓐ 3 Trinomial
    ⓑ 1 Monomial
    ⓒ 5 Polynomial
    ⓓ 2 Binomial
    ⓔ 1 Monomial

    Try It 10.1

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

    1. ⓐ
    2. ⓑ
    3. ⓒ
    4. ⓓ
    5. ⓔ

    Try It 10.2

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

    1. ⓐ
    2. ⓑ
    3. ⓒ
    4. ⓓ
    5. ⓔ

    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.

    A monomial that has no variable, just a constant, is a special case. The degree of a constant is —it has no variable.

    Degree of a Polynomial

    The degree of a term is the exponent of its variable.

    The degree of a constant is

    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.

    Remember: Any base written without an exponent has an implied exponent of

    A table is shown. The top row is titled “Monomials” and lists the following monomials: 5, 4 b squared, negative 9 x cubed, negative 18. The next row is titled “Degree” and lists, in blue, 0, 2, 3, and 0. The next row is titled “Binomial” and lists the following binomials: b plus 1, 3a minus 7, y squared minus 9, 17 x cubed plus 14 x squared. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 1, 0, 1, 0, 2, 0, 3, 2 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 1, 1, 2, 3 in red. The next row is titled “Trinomial” and lists the following trinomials: x squared minus 5x plus 6, 4 y squared minus 7y plus 2, 5 a to the fourth minus 3 a cubed plus a, and x to the fourth plus 2 x squared minus 5. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 2, 1, 0, 2, 1, 0, 4, 3, 1, 4, 2, 0 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 2, 2, 4, 4 in red. The next row is titled “Polynomial” and lists the following polynomials: b plus 1, 4 y squared minus 7y plus 2, and 4 x to the fourth plus x cubed plus 8 x squared minus 9x plus 1. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 1, 0, 2, 1, 0, 4, 3, 2, 1, 0 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 1, 2, 4 in red.

    Example 10.2

    Find the degree of the following polynomials:

    1. ⓐ
    2. ⓑ
    3. ⓒ
    4. ⓓ
    5. ⓔ
    Answer

    ⓐ
    The exponent of is one. The degree is 1.
    ⓑ
    The highest degree of all the terms is 3. The degree is 3
    ⓒ
    The degree of a constant is 0. The degree is 0.
    ⓓ
    The highest degree of all the terms is 2. The degree is 2.
    ⓔ
    The highest degree of all the terms is 1. The degree is 1.

    Try It 10.3

    Find the degree of the following polynomials:

    1. ⓐ
    2. ⓑ
    3. ⓒ
    4. ⓓ
    5. ⓔ

    Try It 10.4

    Find the degree of the following polynomials:

    1. ⓐ
    2. ⓑ
    3. ⓒ
    4. ⓓ
    5. ⓔ

    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. Look back at the polynomials in Example 10.2. Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.

    Add and Subtract Monomials

    In The Language of Algebra, you simplified expressions by combining like terms. Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.

    Example 10.3

    Add:

    Answer

    Combine like terms.

    Try It 10.5

    Add:

    Try It 10.6

    Add:

    Example 10.4

    Subtract:

    Answer

    Combine like terms.

    Try It 10.7

    Subtract:

    Try It 10.8

    Subtract:

    Example 10.5

    Simplify:

    Answer

    Combine like terms.

    Remember, and are not like terms. The variables are not the same.

    Try It 10.9

    Add:

    Try It 10.10

    Add:

    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 with 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 10.6

    Find the sum:

    Answer

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

    Try It 10.11

    Find the sum:

    Try It 10.12

    Find the sum:

    Parentheses are grouping symbols. When we add polynomials as we did in Example 10.6, 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 10.7

    Find the difference:

    Answer

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

    Try It 10.13

    Find the difference:

    Try It 10.14

    Find the difference:

    Example 10.8

    Subtract: from

    Answer

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

    Try It 10.15

    Subtract: from

    Try It 10.16

    Subtract: from

    Evaluate a Polynomial for a Given Value

    In The Language of Algebra we evaluated expressions. Since polynomials are expressions, we'll follow the same procedures to evaluate polynomials—substitute the given value for the variable into the polynomial, and then simplify.

    Example 10.9

    Evaluate when

    1. ⓐ
    2. ⓑ
    Answer

    ⓐ
    Substitute 3 for
    Simplify the expression with the exponent.
    Multiply.
    Simplify.
    ⓑ
    Substitute −1 for
    Simplify the expression with the exponent.
    Multiply.
    Simplify.

    Try It 10.17

    Evaluate: when

    1. ⓐ
    2. ⓑ

    Try It 10.18

    Evaluate: when

    1. ⓐ
    2. ⓑ

    Example 10.10

    The polynomial gives the height of an object seconds after it is dropped from a foot tall bridge. Find the height after seconds.

    Answer

    .
    Substitute 3 for .
    Simplify the expression with the exponent. .
    Multiply. .
    Simplify. .

    Try It 10.19

    The polynomial gives the height, in feet, of a ball seconds after it is tossed into the air, from an initial height of feet. Find the height after seconds.

    Try It 10.20

    The polynomial gives the height, in feet, of a ball seconds after it is tossed into the air, from an initial height of feet. Find the height after seconds.

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    Section 10.1 Exercises

    Practice Makes Perfect

    Identify Polynomials, Monomials, Binomials and Trinomials

    In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial.

    1.

    2.

    3.

    4.

    5.

    6.

    7.

    8.

    Determine the Degree of Polynomials

    In the following exercises, determine the degree of each polynomial.

    9.

    10.

    11.

    12.

    13.

    14.

    Add and Subtract Monomials

    In the following exercises, add or subtract the monomials.

    15.

    16.

    17.

    18.

    19.

    20.

    21.

    Add:

    22.

    Add:

    23.

    24.

    25.

    Subtract

    26.

    Subtract

    Add and Subtract Polynomials

    In the following exercises, add or subtract the polynomials.

    27.

    28.

    29.

    30.

    31.

    32.

    33.

    34.

    35.

    36.

    37.

    38.

    39.

    Find the sum of and

    40.

    Find the sum of and

    41.

    Subtract from

    42.

    Subtract from

    43.

    Find the difference of and

    44.

    Find the difference of and

    Evaluate a Polynomial for a Given Value

    In the following exercises, evaluate each polynomial for the given value.

    45.

    1. ⓐ
    2. ⓑ
    3. ⓒ
    46.

    1. ⓐ
    2. ⓑ
    3. ⓒ
    47.

    1. ⓐ
    2. ⓑ
    3. ⓒ
    48.

    1. ⓐ
    2. ⓑ
    3. ⓒ
    49.

    A window washer drops a squeegee from a platform feet high. The polynomial gives the height of the squeegee seconds after it was dropped. Find the height after seconds.

    50.

    A manufacturer of microwave ovens has found that the revenue received from selling microwaves at a cost of p dollars each is given by the polynomial Find the revenue received when dollars.

    Everyday Math

    51.

    Fuel Efficiency The fuel efficiency (in miles per gallon) of a bus going at a speed of miles per hour is given by the polynomial Find the fuel efficiency when

    52.

    Stopping Distance The number of feet it takes for a car traveling at miles per hour to stop on dry, level concrete is given by the polynomial Find the stopping distance when

    Writing Exercises

    53.

    Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

    54.

    Eloise thinks the sum is What is wrong with her 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.

    This page titled 7.3: Adding and Subtracting Polynomials was last modified on Tue, 15 Aug 2023 18:54:55 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax.