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Mathematics LibreTexts

10.1: Add and Subtract Polynomials

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    5012
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    Skills to Develop

    • 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!

    Before you get started, take this readiness quiz.

    1. Simplify: 8x + 3x. If you missed this problem, review Example 2.22.
    2. Subtract: (5n + 8) − (2n − 1). If you missed this problem, review Example 7.29.
    3. Evaluate: 4y2 when y = 5 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 axm, 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.

    Definition: 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 b + 1 4y2 − 7y + 2 5x5 − 4x4 + x3 + 8x2 − 9x + 1
    Monomial 5 4b2 -9x3
    Binomial 3a - 7 y2 - 9 17x3 + 14x2
    Trinomial x2 - 5x + 6 4y2 - 7y + 2 5a4 - 3a3 + a

    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: (a) 8x2 − 7x − 9 (b) −5a4 (c) x4 − 7x3 − 6x2 + 5x + 2 (d) 11 − 4y3 (e) n

    Solution
    Polynomial Number of terms Type
    (a) 8x2 − 7x − 9 3 Trinomial
    (b) −5a4 1 Monomial
    (c) x4 − 7x3 − 6x2 + 5x + 2 5 Polynomial
    (d) 11 − 4y3 2 Binomial
    (e) n 1 Monomial

    Exercise 10.1:

    Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. (a) z (b) 2x3 − 4x2 − x − 8 (c) 6x2 − 4x + 1 (d) 9 − 4y2 (e) 3x7

    Exercise 10.2:

    Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. (a) y3 − 8 (b) 9x3 − 5x2 − x (c) x4 − 3x2 − 4x − 7 (d) −y4 (e) w

    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 0 —it has no variable.

    Definition: 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.

    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 1.

    CNX_BMath_Figure_10_01_002.jpg

    Example 10.2:

    Find the degree of the following polynomials: (a) 4x (b) 3x3 − 5x + 7 (c) −11 (d) −6x2 + 9x − 3 (e) 8x + 2

    Solution

    (a) 4x

    The exponent of x is one. x = x1 The degree is 1.

    (b) 3x3 − 5x + 7

    The highest degree of all the terms is 3. The degree is 3.

    (c) −11

    The degree of a constant is 0. The degree is 0.

    (d) −6x2 + 9x − 3

    The highest degree of all the terms is 2. The degree is 2.

    (e) 8x + 2

    The highest degree of all the terms is 1. The degree is 1.

    Exercise 10.3:

    Find the degree of the following polynomials: (a) −6y (b) 4x − 1 (c) 3x4 + 4x2 − 8 (d) 2y2 + 3y + 9 (e) −18

    Exercise 10.4:

    Find the degree of the following polynomials: (a) 47 (b) 2x2 − 8x + 2 (c) x4 − 16 (d) y5 − 5y3 + y (e) 9a3

    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: 17x2 + 6x2.

    Solution
    Combine like terms. 23x2

    Exercise 10.5:

    Add: 12x2 + 5x2.

    Exercise 10.6:

    Add: −11y+ 8y2.

    Example 10.4:

    Subtract: 11n − (−8n).

    Solution
    Combine like terms. 19n

    Exercise 10.7:

    Subtract: 9n − (−5n).

    Exercise 10.8:

    Subtract: −7a3 − (−5a3).

    Example 10.5:

    Simplify: a2 + 4b2 − 7a2.

    Solution
    Combine like terms. −6a2 + 4b2

     Remember, −6a2 and 4b2 are not like terms. The variables are not the same.

    Exercise 10.9:

    Add: 3x2 + 3y2 − 5x2.

    Exercise 10.10:

    Add: 2a2 + b2 − 4a2.

    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: (4x2 − 5x + 1) + (3x2 − 8x − 9)

    Solution
    Identify like terms. CNX_BMath_Figure_10_01_003-02.png
    Rearrange to get the like terms together. CNX_BMath_Figure_10_01_003_img-03.png
    Combine like terms. CNX_BMath_Figure_10_01_003_img-04.png

    Exercise 10.11:

    Find the sum: (3x2 − 2x + 8) + (x2 − 6x + 2).

    Exercise 10.12:

    Find the sum: (7y2 + 4y − 6) + (4y2 + 5y + 1)

    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: (7u2 − 5u + 3) − (4u2 − 2).

    Solution
    Distribute and identify like terms. CNX_BMath_Figure_10_01_004_img-02.png
    Rearrange the terms. CNX_BMath_Figure_10_01_004_img-03.png
    Combine like terms. CNX_BMath_Figure_10_01_004_img-04.png

    Exercise 10.13:

    Find the difference: (6y2 + 3y − 1) − (3y2 − 4).

    Exercise 10.14:

    Find the difference: (8u2 − 7u − 2) − (5u2 − 6u − 4).

    Example 10.8:

    Subtract: (m2 − 3m + 8) from (9m2 − 7m + 4).

    Solution
    Distribute and identify like terms. CNX_BMath_Figure_10_01_005_img-02.png
    Rearrange the terms. CNX_BMath_Figure_10_01_005_img-03.png
    Combine like terms. CNX_BMath_Figure_10_01_005_img-04.png

    Exercise 10.15:

    Subtract: (4n2 − 7n − 3) from (8n2 + 5n − 3).

    Exercise 10.16:

    Subtract: (a2 − 4a − 9) from (6a2 + 4a − 1).

    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 3x 2 − 9x + 7 when (a) x = 3 (b) x = −1

    Solution

    (a) x = 3

    Substitute 3 for x. 3(3)2 − 9(3) + 7
    Simplify the expression with the exponent. 3 • 9 − 9(3) + 7
    Multiply. 27 − 27 + 7
    Simplify. 7

    (b) x = −1

    Substitute -1 for x. 3(-1)2 − 9(-1) + 7
    Simplify the expression with the exponent. 3 • 1 − 9(-1) + 7
    Multiply. 3 + 9 + 7
    Simplify. 19

    Exercise 10.17:

    Evaluate: 2x2 + 4x − 3 when (a) x = 2 (b) x = −3

    Exercise 10.18:

    Evaluate: 7y2 − y − 2 when (a) y = −4 (b) y = 0

    Example 10.10:

    The polynomial −16t 2 + 300 gives the height of an object t seconds after it is dropped from a 300 foot tall bridge. Find the height after t = 3 seconds.

    Solution
    Substitute 3 for t. -16(3)2 + 300
    Simplify the expression with the exponent. -16 • 9 + 300
    Multiply. -144 + 300
    Simplify. 156

    Exercise 10.19:

    The polynomial −8t2 + 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.

    Exercise 10.20:

    The polynomial −8t2 + 24t + 4 gives the height, in feet, of a ball x seconds after it is tossed into the air, from an initial height of 4 feet. Find the height after t = 2 seconds.

    ACCESS ADDITIONAL ONLINE RESOURCES

    Adding Polynomials  

    Subtracting Polynomials

    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. 5x + 2
    2. z2 − 5z − 6
    3. a2 + 9a + 18
    4. −12p4
    5. y3 − 8y2 + 2y − 16
    6. 10 − 9x
    7. 23y2
    8. m4 + 4m3 + 6m2 + 4m + 1

    Determine the Degree of Polynomials

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

    1. 8a5 − 2a3 + 1
    2. 5c3 + 11c2 − c − 8
    3. 3x − 12
    4. 4y + 17
    5. −13
    6. −22

    Add and Subtract Monomials

    In the following exercises, add or subtract the monomials.

    1. 6x2 + 9x2
    2. 4y3 + 6y3
    3. −12u + 4u
    4. −3m + 9m
    5. 5a + 7b
    6. 8y + 6z
    7. Add: 4a, − 3b, − 8a
    8. Add: 4x, 3y, − 3x
    9. 18x − 2x
    10. 13a − 3a
    11. Subtract 5x6 from − 12x6
    12. Subtract 2p4 from − 7p4

    Add and Subtract Polynomials

    In the following exercises, add or subtract the polynomials.

    1. (4y2 + 10y + 3) + (8y2 − 6y + 5)
    2. (7x2 − 9x + 2) + (6x2 − 4x + 3)
    3. (x2 + 6x + 8) + (−4x2 + 11x − 9)
    4. (y2 + 9y + 4) + (−2y2 − 5y − 1)
    5. (3a2 + 7) + (a2 − 7a − 18)
    6. (p2 − 5p − 11) + (3p2 + 9)
    7. (6m2 − 9m − 3) − (2m2 + m − 5)
    8. (3n2 − 4n + 1) − (4n2 − n − 2)
    9. (z2 + 8z + 9) − (z2 − 3z + 1)
    10. (z2 − 7z + 5) − (z2 − 8z + 6)
    11. (12s2 − 15s) − (s − 9)
    12. (10r2 − 20r) − (r − 8)
    13. Find the sum of (2p3 − 8) and (p2 + 9p + 18)
    14. Find the sum of (q2 + 4q + 13) and (7q3 − 3)
    15. Subtract (7x2 − 4x + 2) from (8x2 − x + 6) 
    16. Subtract (5x2 − x + 12) from (9x2 − 6x − 20)
    17. Find the difference of (w2 + w − 42) and (w2 − 10w + 24)
    18. Find the difference of (z2 − 3z − 18) and (z2 + 5z − 20)

    Evaluate a Polynomial for a Given Value

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

    1. Evaluate 8y2 − 3y + 2
      1. y = 5
      2. y = −2
      3. y = 0
    2. Evaluate 5y2 − y − 7 when:
      1. y = −4
      2. y = 1
      3. y = 0
    3. Evaluate 4 − 36x when:
      1. x = 3
      2. x = 0
      3. x = −1
    4. Evaluate 16 − 36x2 when:
      1. x = −1
      2. x = 0
      3. x = 2
    5. A window washer drops a squeegee from a platform 275 feet high. The polynomial −16t2 + 275 gives the height of the squeegee t seconds after it was dropped. Find the height after t = 4 seconds.
    6. 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 −5p2 + 350p. Find the revenue received when p = 50 dollars.

    Everyday Math

    1. Fuel Efficiency The fuel efficiency (in miles per gallon) of a bus going at a speed of x miles per hour is given by the polynomial \(− \frac{1}{160} x^{2} + \frac{1}{2} x\). Find the fuel efficiency when x = 40 mph.
    2. 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 0.06x2 + 1.1x. Find the stopping distance when x = 60 mph.

    Writing Exercises

    1. Using your own words, explain the difference between a monomial, a binomial, and a trinomial.
    2. Eloise thinks the sum 5x2 + 3x4 is 8x6. What is wrong with her reasoning?

    Self Check

    (a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    CNX_BMath_Figure_AppB_058.2.jpg

    (b) 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.

    Who 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.

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