Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Algebra 1

6.1.2 Adding and Subtracting Polynomials

Algebra 16.1.2 Adding and Subtracting Polynomials
Search for key terms or text.

Activity

We will solve these two problems as a class.

1. Do these terms have the same variables with the same exponents? Explain your answer.

6 x 4 y 2 6 x 4 y 2           8 x 4 y 2 8 x 4 y 2

2. Add the terms together: 6 x 4 y 2 + 8 x 4 y 2 6 x 4 y 2 + 8 x 4 y 2 . What is the sum?

In problems 3 - 7, simplify the monomial addition and subtraction problems on your own.

3. 12 q + 9 q 12 q + 9 q

4. − 14 c 3 + 5 c 3 − 14 c 3 + 5 c 3

5. 8 g 2 h 2 + ( − 6 g 2 h 2 ) 8 g 2 h 2 + ( − 6 g 2 h 2 )

6. 8 m n 3 p − ( − 5 m n 3 p ) 8 m n 3 p − ( − 5 m n 3 p )

7. − 16 a 2 + 3 b 2 + 8 a 2 − 16 a 2 + 3 b 2 + 8 a 2

8. Can the following monomials be simplified using like terms? Explain. 17 a 2 b c 3 + 6 a 2 b 2 c 3 17 a 2 b c 3 + 6 a 2 b 2 c 3

In problems 9 - 14, simplify the polynomial addition and subtraction problems on your own.

9. ( 5 k + 7 m − 2 ) + ( 7 k + 9 m − 4 ) ( 5 k + 7 m − 2 ) + ( 7 k + 9 m − 4 )

10. ( 12 w − 11 ) − ( 5 w + 6 ) ( 12 w − 11 ) − ( 5 w + 6 )

11. ( 5 k 2 + 7 k − 2 ) + ( 7 k 2 + 9 k − 4 ) ( 5 k 2 + 7 k − 2 ) + ( 7 k 2 + 9 k − 4 )

You can use the following video if you need some help.

Access multimedia content

12. ( 9 h 2 − 4 h + 17 ) + ( 2 h 2 − 9 ) ( 9 h 2 − 4 h + 17 ) + ( 2 h 2 − 9 )

13. ( 8 f 2 + 6 f v − 3 v 2 ) − ( 6 f 2 − 3 v 2 ) ( 8 f 2 + 6 f v − 3 v 2 ) − ( 6 f 2 − 3 v 2 )

14. ( 8 z 3 − 6 z x + 3 x 2 ) + ( 12 z 3 − 4 z x ) ( 8 z 3 − 6 z x + 3 x 2 ) + ( 12 z 3 − 4 z x )

Are you ready for more?

Extending Your Thinking

Simplify.

( 6 a 3 − 2 a 2 b ) − 2 ( a b 2 + 3 b 3 ) + ( 2 a 2 b + 2 a b 2 ) ( 6 a 3 − 2 a 2 b ) − 2 ( a b 2 + 3 b 3 ) + ( 2 a 2 b + 2 a b 2 )

Self Check

Find the sum.

( 14 y 2 + 6 y − 4 ) + ( 3 y 2 + 8 y + 5 )

  1. 11 y 2 + 14 y + 1
  2. 17 y 2 + 14 y − 9
  3. 11 y 2 − 2 y − 9
  4. 17 y 2 + 14 y + 1

Additional Resources

Adding and Subtracting Monomials

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

These terms have the same variables and the same exponents:

  • 5 m 3 n 2 5 m 3 n 2           2 m 3 n 2 2 m 3 n 2

So we can add the monomials:

  • 5 m 3 n 2 + 2 m 3 n 2 = 7 m 3 n 2 5 m 3 n 2 + 2 m 3 n 2 = 7 m 3 n 2

The variables and exponents of the monomials do not change when we add or subtract them.

Example 1

18 a + 2 a 18 a + 2 a

Step 1 - Combine like terms.

18 a + 2 a = 20 a 18 a + 2 a = 20 a

Example 2

37 t − 15 t 37 t − 15 t

Step 1 - Combine like terms.

37 t − 15 t = 22 t 37 t − 15 t = 22 t

Example 3

25 y 2 + 15 y 2 25 y 2 + 15 y 2

Step 1 - Combine like terms.

25 y 2 + 15 y 2 = 40 y 2 25 y 2 + 15 y 2 = 40 y 2

Try it

Try It: Adding and Subtracting Monomials

16 p q 3 − ( − 7 p q 3 ) 16 p q 3 − ( − 7 p q 3 )

Adding and Subtracting 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.

Let’s look at some examples.

Example 1

Find the sum: ( 7 y 2 − 2 y + 9 ) + ( 4 y 2 − 8 y − 7 ) ( 7 y 2 − 2 y + 9 ) + ( 4 y 2 − 8 y − 7 ) .

Step 1 - Identify like terms by finding those with the same variable and same exponent.

( 7 y 2 − 2 y + 9 ) + ( 4 y 2 − 8 y − 7 ) ( 7 y 2 − 2 y + 9 ) + ( 4 y 2 − 8 y − 7 )

Step 2 - Use the Commutative Property to rearrange like terms together. Be sure to take the appropriate sign with each term ( + / − ) ( + / − ) .

7 y 2 + 4 y 2 − 2 y − 8 y + 9 − 7 7 y 2 + 4 y 2 − 2 y − 8 y + 9 − 7

Step 3 - Combine like terms.

1 1 y 2 − 1 0 y + 2 1 1 y 2 − 1 0 y + 2

Example 2

Find the sum: ( 22 g − 14 h + 3 ) + ( 10 g + 8 h − 9 ) ( 22 g − 14 h + 3 ) + ( 10 g + 8 h − 9 )

Step 1 - Identify like terms.

( 2 2 g − 1 4 h + 3 ) + ( 1 0 g + 8 h − 9 ) ( 2 2 g − 1 4 h + 3 ) + ( 1 0 g + 8 h − 9 )

Color 22g, 10g in red.  14h, 8h in blue.

Step 2 - Use the Commutative Property to rearrange like terms together. Be sure to take the appropriate sign with each term (+/−).

( 2 2 g + 1 0 g − 1 4 h + 8 h + 3 + 9 ) ( 2 2 g + 1 0 g − 1 4 h + 8 h + 3 + 9 )

Step 3 - Combine like terms.

32 g − 6 h + 12 32 g − 6 h + 12

Example 3

Find the sum: ( 8 j + 7 k ) + ( 2 s 2 − 3 k + 26 ) ( 8 j + 7 k ) + ( 2 s 2 − 3 k + 26 )

Step 1 - Identify like terms.

( 8 j + 7 k − 3 ) + ( 2 s 2 − 3 k + 26 ) ( 8 j + 7 k − 3 ) + ( 2 s 2 − 3 k + 26 )

Step 2 - Rearrange like terms together.

2 s 2 + 7 k − 3 k + 8 j − 3 + 26 2 s 2 + 7 k − 3 k + 8 j − 3 + 26

Step 3 - Combine like terms.

Example 4

Find the difference: ( 15 g − 4 h + 8 ) − ( 10 g + 13 h − 1 ) ( 15 g − 4 h + 8 ) − ( 10 g + 13 h − 1 )

Step 1 - Identify like terms.

( 1 5 g − 4 h + 8 ) − ( 1 0 g + 1 3 h − 1 ) ( 1 5 g − 4 h + 8 ) − ( 1 0 g + 1 3 h − 1 )

Step 2 - Rearrange like terms together.

Remember to distribute the subtraction to all terms in the second polymnomial.

1 5 g − 1 0 g − 4 h − 1 3 h + 8 + 1 1 5 g − 1 0 g − 4 h − 1 3 h + 8 + 1

Step 3 - Combine like terms.

5 g − 17 h + 9 5 g − 17 h + 9

Example 5

Find the difference: ( 16 y 2 + 12 y + 9 ) − ( 3 y 2 + 2 y − 12 ) ( 16 y 2 + 12 y + 9 ) − ( 3 y 2 + 2 y − 12 )

Step 1 - Identify like terms.

( 1 6 y 2 + 1 2 y + 9 ) − ( 3 y 2 + 2 y − 12 ) ( 1 6 y 2 + 1 2 y + 9 ) − ( 3 y 2 + 2 y − 12 )

Step 2 - Rearrange like terms together.

( 1 6 y 2 − 3 y 2 + 1 2 y − 2 y + 9 + 12 ) ( 1 6 y 2 − 3 y 2 + 1 2 y − 2 y + 9 + 12 )

Step 3 - Combine like terms.

( 13 y 2 + 10 y + 21 ) ( 13 y 2 + 10 y + 21 )

Try it

Try It: Adding and Subtracting Polynomials

Find the sum and difference. If necessary, be careful with the signs as you distribute while subtracting the polynomials.

1. ( 7 x 2 − 4 x + 5 ) + ( x 2 − 7 x + 3 ) ( 7 x 2 − 4 x + 5 ) + ( x 2 − 7 x + 3 )

2. ( 9 w 2 − 7 w + 5 ) − ( 2 w 2 − 4 ) ( 9 w 2 − 7 w + 5 ) − ( 2 w 2 − 4 )

3. ( 8 a + 12 b ) − ( 10 a + 3 b ) ( 8 a + 12 b ) − ( 10 a + 3 b )

Write down your answer, then select the solution button to compare your work.

PreviousNext
Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

© May 21, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.