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Mathematics LibreTexts

6.1: Add and Subtract 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
Quiz

Before you get started, take this readiness quiz.

  1. Simplify: 8𝑥 +3𝑥.
    If you missed this problem, review Example 1.3.37.
  2. Subtract: (5𝑛 +8) (2𝑛 1).
    If you missed this problem, review Example 1.10.52.
  3. Write in expanded form: 𝑎5.
    If you missed this problem, review Example 1.3.7.

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 𝑎𝑥𝑚, where 𝑎 is a constant and 𝑚 is a whole number, it is called a monomial. Some examples of monomial are 8,2𝑥2,4𝑦3, and 11𝑧7.

Definition: Monomials

A monomial is a term of the form 𝑎𝑥𝑚, 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.

Definitions: 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.

 Polynomial 𝑏+14𝑦27𝑦+24𝑥4+𝑥3+8𝑥29𝑥+1 Monomial 148𝑦29𝑥3𝑦513 Binomial 𝑎+74𝑏5𝑦2163𝑥39𝑥2 Trinomial 𝑥27𝑥+129𝑦2+2𝑦86𝑚4𝑚3+8𝑚𝑧4+3𝑧21

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 6.1.1

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

  1. 4𝑦2 8𝑦 6
  2. 5𝑎4𝑏2
  3. 2𝑥5 5𝑥3 3𝑥 +4
  4. 13 5𝑚3
  5. q

Solution

 Polynomial  Number of terms  Type  (a) 4𝑦28𝑦63 Trinomial  (b) 5𝑎4𝑏21 Monomial  (c) 2𝑥55𝑥39𝑥2+3𝑥+45 Ponomial  (d) 135𝑚32 Binomial  (e) 𝑞1 Monomial 

Try It 6.1.2

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

  1. 5b
  2. 8𝑦3 7𝑦2 𝑦 3
  3. 3𝑥2 5𝑥 +9
  4. 81 4𝑎2
  5. 5𝑥6
Answer
  1. monomial
  2. polynomial
  3. trinomial
  4. binomial
  5. monomial
Try It 6.1.3

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

  1. 27𝑧3 8
  2. 12𝑚3 5𝑚2 2𝑚
  3. 56
  4. 8𝑥4 7𝑥2 6𝑥 5
  5. 𝑛4
Answer
  1. binomial
  2. trinomial
  3. monomial
  4. polynomial
  5. monomial

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, i.e., it has no variable.

Definition: 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.


This table has 11 rows and 5 columns. The first column is a header column, and it names each row. The first row is named “Monomial,” and each cell in this row contains a different monomial. The second row is named “Degree,” and each cell in this row contains the degree of the monomial above it. The degree of 14 is 0, the degree of 8y squared is 2, the degree of negative 9x cubed y to the fifth power is 8, and the degree of negative 13a is 1. The third row is named “Binomial,” and each cell in this row contains a different binomial. The fourth row is named “Degree of each term,” and each cell contains the degrees of the two terms in the binomial above it. The fifth row is named “Degree of polynomial,” and each cell contains the degree of the binomial as a whole.” The degrees of the terms in a plus 7 are 0 and 1, and the degree of the whole binomial is 1. The degrees of the terms in 4b squared minus 5b are 2 and 1, and the degree of the whole binomial is 2. The degrees of the terms in x squared y squared minus 16 are 4 and 0, and the degree of the whole binomial is 4. The degrees of the terms in 3n cubed minus 9n squared are 3 and 2, and the degree of the whole binomial is 3. The sixth row is named “Trinomial,” and each cell in this row contains a different trinomial. The seventh row is named “Degree of each term,” and each cell contains the degrees of the three terms in the trinomial above it. The eighth row is named “Degree of polynomial,” and each cell contains the degree of the trinomial as a whole. The degrees of the terms in x squared minus 7x plus 12 are 2, 1, and 0, and the degree of the whole trinomial is 2. The degrees of the terms in 9a squared plus 6ab plus b squared are 2, 2, and 2, and the degree of the trinomial as a whole is 2. The degrees of the terms in 6m to the fourth power minus m cubed n squared plus 8mn to the fifth power are 4, 5, and 6, and the degree of the whole trinomial is 6. The degrees of the terms in z to the fourth power plus 3z squared minus 1 are 4, 2, and 0, and the degree of the whole trinomial is 4. The ninth row is named “Polynomial,” and each cell contains a different polynomial. The tenth row is named “Degree of each term,” and the eleventh row is named “Degree of polynomial.” The degrees of the terms in b plus 1 are 1 and 0, and the degree of the whole polynomial is 1. The degrees of the terms in 4y squared minus 7y plus 2 are 2, 1, and 0, and the degree of the whole polynomial is 2. The degrees of the terms in 4x to the fourth power plus x cubed plus 8x squared minus 9x plus 1 are 4, 3, 2, 1, and 0, and the degree of the whole polynomial is 4.

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 6.1.4

Find the degree of the following polynomials.

  1. 10y
  2. 4𝑥3 7𝑥 +5
  3. −15
  4. 8𝑏2 +9𝑏 2
  5. 8𝑥𝑦2 +2𝑦

Solution

  1. 10𝑦The exponent of y is one. 𝑦=𝑦1The degree is 1.
  2. 4𝑥37𝑥+5The highest degree of all the terms is 3.The degree is 3.
  3. 15The degree of a constant is 0.The degree is 0.
  4. 8𝑏2+9𝑏2The highest degree of all the terms is 2.The degree is 2.
  5. 8𝑥𝑦2+2𝑦The highest degree of all the terms is 3.The degree is 3.
Try It 6.1.5

Find the degree of the following polynomials:

  1. −15b
  2. 10𝑧4 +4𝑧2 5
  3. 12𝑐5𝑑4 +9𝑐3𝑑9 7
  4. 3𝑥2𝑦 4𝑥
  5. −9
Answer
  1. 1
  2. 4
  3. 12
  4. 3
  5. 0
Try It 6.1.6

Find the degree of the following polynomials:

  1. 52
  2. 𝑎4𝑏 17𝑎4
  3. 5𝑥 +6𝑦 +2𝑧
  4. 3𝑥2 5𝑥 +7
  5. 𝑎3
Answer
  1. 0
  2. 5
  3. 1
  4. 2
  5. 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 6.1.7

Add:25𝑦2 +15𝑦2

Solution

25𝑦2+15𝑦2Combine like terms.40𝑦2

Try It 6.1.8

Add: 12𝑞2 +9𝑞2

Answer

21𝑞2

Try It 6.1.9

Add:15𝑐2 +8𝑐2

Answer

7𝑐2

Example 6.1.10

Subtract: 16p−(−7p)

Solution

16𝑝(7𝑝)Combine like terms.23𝑝

Try It 6.1.11

Subtract: 8m−(−5m).

Answer

13m

Try It 6.1.12

Subtract: 15𝑧3 (5𝑧3)

Answer

10𝑧3

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

Example 6.1.13

Simplify: 𝑐2 +7𝑑2 6𝑐2

Solution

𝑐2+7𝑑26𝑐2Combine like terms.5𝑐2+7𝑑2

Try It 6.1.14

Add: 8𝑦2 +3𝑧2 3𝑦2

Answer

5𝑦2 +3𝑧2

Try It 6.1.15

Add: 3𝑚2 +𝑛2 7𝑚2

Answer

4𝑚2 +𝑛2

Example 6.1.16

Simplify: 𝑢2𝑣 +5𝑢2 3𝑣2

Solution

𝑢2𝑣+5𝑢23𝑣2There are no like terms to combine.𝑢2𝑣+5𝑢23𝑣2

Try It 6.1.17

Simplify: 𝑚2𝑛2 8𝑚2 +4𝑛2

Answer

There are no like terms to combine.

Try It 6.1.18

Simplify: 𝑝𝑞2 6𝑝 5𝑞2

Answer

There are no like terms to combine.

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 6.1.19

Find the sum: (5𝑦23𝑦+15) +(3𝑦24𝑦11)

Solution

Identify like terms. 5 y squared minus 3 y plus 15, plus 3 y squared minus 4 y minus 11.
Rearrange to get the like terms together. 5y squared plus 3y squared, identified as like terms, minus 3y minus 4y, identified as like terms, plus 15 minus 11, identified as like terms.
Combine like terms. 8 y squared minus 7y plus 4.
Try It 6.1.20

Find the sum: (7𝑥24𝑥+5) +(𝑥27𝑥+3)

Answer

8𝑥2 11𝑥 +1

Try It 6.1.21

Find the sum:(14𝑦2+6𝑦4) +(3𝑦2+8𝑦+5)

Answer

17𝑦2 +14𝑦 +1

Example 6.1.22

Find the difference: (9𝑤27𝑤+5) (2𝑤24)

Solution

  9 w squared minus 7 w plus 5, minus 2 w squared minus 4.
Distribute and identify like terms. 9 w squared and 2 w squared are like terms. 5 and 4 are also like terms.
Rearrange the terms. 9 w squared minus 2 w squared minus 7 w plus 5 plus 4.
Combine like terms. 7 w squared minus 7 w plus 9.
Try It 6.1.23

Find the difference: (8𝑥2+3𝑥19) (7𝑥214)

Answer

15𝑥2 +3𝑥 5

Try It 6.1.24

Find the difference: (9𝑏25𝑏4) (3𝑏25𝑏7)

Answer

6𝑏2 +3

Example 6.1.25

Subtract: (𝑐24𝑐+7) from (7𝑐25𝑐+3)

Solution

  .
  7 c squared minus 5 c plus 3, minus c squared minus 4c plus 7.
Distribute and identify like terms. 7 c squared and c squared are like terms. Minus 5c and 4c are like terms. 3 and minus 7 are like terms.
Rearrange the terms. 7 c squared minus c squared minus 5 c plus 4 c plus 3 minus 7.
Combine like terms. 6 c squared minus c minus 4.
Try It 6.1.26

Subtract: (5𝑧26𝑧2) from (7𝑧2+6𝑧4)

Answer

2𝑧2 +12𝑧 2

Try It 6.1.27

Subtract: (𝑥25𝑥8) from (6𝑥2+9𝑥1)

Answer

5𝑥2 +14𝑥 +7

Example 6.1.28

Find the sum: (𝑢26𝑢𝑣+5𝑣2) +(3𝑢2+2𝑢𝑣)

Solution

(𝑢26𝑢𝑣+5𝑣2)+(3𝑢2+2𝑢𝑣)Distribute.𝑢26𝑢𝑣+5𝑣2+3𝑢2+2𝑢𝑣Rearrange the terms, to put like terms together𝑢2+3𝑢26𝑢𝑣+2𝑢𝑣+5𝑣2Combine like terms.4𝑢24𝑢𝑣+5𝑣2

Try It 6.1.29

Find the sum: (3𝑥24𝑥𝑦+5𝑦2) +(2𝑥2𝑥𝑦)

Answer

5𝑥2 5𝑥𝑦 +5𝑦2

Try It 6.1.30

Find the sum: (2𝑥23𝑥𝑦2𝑦2) +(5𝑥23𝑥𝑦)

Answer

7𝑥2 6𝑥𝑦 2𝑦2

Example 6.1.31

Find the difference: (𝑝2+𝑞2) (𝑝2+10𝑝𝑞2𝑞2)

Solution

(𝑝2+𝑞2)(𝑝2+10𝑝𝑞2𝑞2)Distribute.𝑝2+𝑞2𝑝210𝑝𝑞+2𝑞2Rearrange the terms, to put like terms together𝑝2𝑝210𝑝𝑞+𝑞2+2𝑞2Combine like terms.10𝑝𝑞+3𝑞2

Try It 6.1.32

Find the difference: (𝑎2+𝑏2) (𝑎2+5𝑎𝑏6𝑏2)

Answer

5𝑎𝑏 5𝑏2

Try It 6.1.33

Find the difference: (𝑚2+𝑛2) (𝑚27𝑚𝑛3𝑛2)

Answer

4𝑛2 +7𝑚𝑛

Example 6.1.34

Simplify: (𝑎3𝑎2𝑏) (𝑎𝑏2+𝑏3) +(𝑎2𝑏+𝑎𝑏2)

Solution

(𝑎3𝑎2𝑏)(𝑎𝑏2+𝑏3)+(𝑎2𝑏+𝑎𝑏2)Distribute.𝑎3𝑎2𝑏𝑎𝑏2𝑏3+𝑎2𝑏+𝑎𝑏2Rearrange the terms, to put like terms together𝑎3𝑎2𝑏+𝑎2𝑏𝑎𝑏2+𝑎𝑏2𝑏3Combine like terms.𝑎3𝑏3

Try It 6.1.35

Simplify: (𝑥3𝑥2𝑦) (𝑥𝑦2+𝑦3) +(𝑥2𝑦+𝑥𝑦2)

Answer

𝑥3 𝑦3

Try It 6.1.36

Simplify: (𝑝3𝑝2𝑞) +(𝑝𝑞2+𝑞3) (𝑝2𝑞+𝑝𝑞2)

Answer

𝑝3 2𝑝2𝑞 +𝑞3

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 6.1.37

Evaluate 5𝑥2 8𝑥 +4 when

  1. x=4
  2. x=−2
  3. x=0

Solution

1. x=4  
  5 x squared minus 8 x plus 4.
Substitute 4 for x. 5 times 4 squared minus 8 times 4 plus 4.
Simplify the exponents. 5 times 16 minus 8 times 4 plus 4.
Multiply. 80 minus 32 plus 4.
Simplify. 52.
2. x=−2  
  5 x squared minus 8 x plus 4.
Substitute negative 2 for x. 5 times negative 2 squared minus 8 times negative 2 plus 4.
Simplify the exponents. 5 times 4 minus 8 times negative 2 plus 4.
Multiply. 20 plus 16 plus 4.
Simplify. 40.
3. x=0  
  5 x squared minus 8 x plus 4.
Substitute 0 for x. 5 times 0 squared minus 8 times 0 plus 4.
Simplify the exponents. 5 times 0 minus 8 times 0 plus 4.
Multiply. 0 plus 0 plus 4.
Simplify. 4.
Try It 6.1.38

Evaluate: 3𝑥2 +2𝑥 15 when

  1. x=3
  2. x=−5
  3. x=0
Answer
  1. 18
  2. 50
  3. −15
Try It 6.1.39

Evaluate: 5𝑧2 𝑧 4 when

  1. z=−2
  2. z=0
  3. z=2
Answer
  1. 18
  2. −4
  3. 14
Example 6.1.40

The polynomial 16𝑡2 +250 gives the height of a ball tt seconds after it is dropped from a 250 foot tall building. Find the height after t=2 seconds.

Solution

16𝑡2+250Substitute t = 2.16(2)2+250Simplify 164+250Simplify 64+250Simplify 186After 2 seconds the height of the ball is 186 feet. 

Try It 6.1.41

The polynomial 16𝑡2 +250 gives the height of a ball tt seconds after it is dropped from a 250 foot tall building. Find the height after t=0 seconds.

Answer

250

Try It 6.1.42

The polynomial 16𝑡2 +250 gives the height of a ball tt seconds after it is dropped from a 250 foot tall building. Find the height after t=3 seconds.

Answer

106

Example 6.1.43

The polynomial 6𝑥2 +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=6y=6 feet.

Solution

  6 x squared plus 15 x y.
Substitute x equals 4 and y equals 6. 6 times 4 squared plus 15 times 4 times 6.
Simplify. 6 times 16 plus 15 times 4 times 6.
Simplify. 96 plus 360.
Simplify. 456.
  The cost of producing the box is $456.
Try It 6.1.43

The polynomial 6𝑥2 +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.

Answer

$576

Try It 6.1.44

The polynomial 6𝑥2 +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.

Answer

$750

Key Concepts

  • Monomials
    • A monomial is a term of the form 𝑎𝑥𝑚, where aa is a constant and mm is a whole number
  • 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.
  • 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 exponent of its variable.
monomial
A monomial is a term of the form 𝑎𝑥𝑚, where a is a constant and m is a whole number; a monomial has exactly one term.
polynomial
A polynomial is a monomial, or two or more monomials combined by addition or subtraction.
standard form
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.

This page titled 6.1: Add and Subtract Polynomials is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by OpenStax.

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