Skip to ContentGo to accessibility page
OpenStax Logo
Algebra 1

6.2.3 Multiplying a Polynomial by a Polynomial

Algebra 16.2.3 Multiplying a Polynomial by a Polynomial

6.2.3 Multiplying a Polynomial by a Polynomial

6.2.3 • Multiplying a Polynomial by a Polynomial

Activity

For questions 1 – 5, match each polynomial expression to its simplified solution. Be prepared to explain or show your work.

1. ( c + 1 ) ( c 2 + 3 c + 6 ) ( c + 1 ) ( c 2 + 3 c + 6 )

2. ( c − 6 ) ( c 2 + c + 1 ) ( c − 6 ) ( c 2 + c + 1 )

3. ( c + 5 ) ( c 2 + 2 c − 4 ) ( c + 5 ) ( c 2 + 2 c − 4 )

4. ( c 2 − 3 ) ( c 2 − 4 c − 1 ) ( c 2 − 3 ) ( c 2 − 4 c − 1 )

5. ( c 2 + 10 ) ( c 2 − 5 c + 2 ) ( c 2 + 10 ) ( c 2 − 5 c + 2 )

6. Now, fill in the missing blanks to complete the polynomial multiplication. ( 2 x − 4 ) ( x 2 + ◻ x − 5 ) = 2 x ( x 2 + 6 x − 5 ) − 4 ( x 2 + 6 x − ◻ ) = 2 x 3 + ◻ x 2 − 10 x − 4 x 2 − 24 x + ◻ = ◻ x 3 + 8 x 2 − ◻ x + 20 ( 2 x − 4 ) ( x 2 + ◻ x − 5 ) = 2 x ( x 2 + 6 x − 5 ) − 4 ( x 2 + 6 x − ◻ ) = 2 x 3 + ◻ x 2 − 10 x − 4 x 2 − 24 x + ◻ = ◻ x 3 + 8 x 2 − ◻ x + 20

Video: Multiplying Polynomials Using Different Methods

Watch the following video to learn more about how to multiply polynomials using both the Distributive Property and Vertical Alignment.

Multiplying Polynomials Using Different Methods

Self Check

Multiply.

( d + 4 ) ( 2 d 2 − 3 d + 5 )

  1. 8 d 2 − 12 d + 20
  2. 2 d 3 − 3 d 2 + 5 d
  3. 2 d 3 + 5 d 2 − 7 d + 20
  4. 2 d 3 − 11 d 2 − 17 d + 20

Additional Resources

Multiplying a Polynomial by a Polynomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use other representations of the Distributive Property such as Vertical Alignment or distributing individual terms or polynomials.

Example 1

Multiply ( b + 3 ) ( 2 b 2 − 5 b + 8 ) ( b + 3 ) ( 2 b 2 − 5 b + 8 ) using the Distributive Property by distributing either a binomial or an individual term.

Step 1 - Distribute.

( 2 b 2 − 5 b + 8 ) ( 2 b 2 − 5 b + 8 ) .

b ( 2 b 2 − 5 b + 8 ) + 3 ( 2 b 2 − 5 b + 8 ) b ( 2 b 2 − 5 b + 8 ) + 3 ( 2 b 2 − 5 b + 8 )

Step 2 - Multiply.

2 b 3 − 5 b 2 + 8 b + 6 b 2 − 15 b + 24 2 b 3 − 5 b 2 + 8 b + 6 b 2 − 15 b + 24

Step 3 - Combine like terms.

2 b 3 + b 2 − 7 b + 24 2 b 3 + b 2 − 7 b + 24

Example 2

Multiply ( b + 3 ) ( 2 b 2 − 5 b + 8 ) ( b + 3 ) ( 2 b 2 − 5 b + 8 ) using vertical alignment.

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Step 1 - Multiply.

( 2 b 2 − 5 b + 8 ) ( 2 b 2 − 5 b + 8 ) by 3 3 .

Step 2 - Multiply.

( 2 b 2 − 5 b + 8 ) ( 2 b 2 − 5 b + 8 ) by b b .

Step 3 - Add like terms.

2 b 2 − 5 b + 8 × b + 3 ― 6 b 2 − 15 b + 24 2 b 3 − 5 b 2 + 8 b ― 2 b 3 + b 2 − 7 b + 24 2 b 2 − 5 b + 8 × b + 3 ― 6 b 2 − 15 b + 24 2 b 3 − 5 b 2 + 8 b ― 2 b 3 + b 2 − 7 b + 24

Try it

Multiplying a Polynomial by a Polynomial

For questions 1 – 2, multiply ( z − 3 ) ( z 2 − 5 z + 2 ) ( z − 3 ) ( z 2 − 5 z + 2 ) using the listed method.

1. The Distributive Property

2. Vertical Alignment

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

PreviousNext
Order a print copy
Citation/Attribution
Reuse and redistribution of this content in digital or print format:
  • 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 prior written permission.
  • This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License, which means that you can reuse and modify the material only for noncommercial purposes, must attribute OpenStax, and must distribute any derivative works under the same license.
  • Any commercial printing of this textbook, including using a local or custom printer, must be approved by OpenStax, and proper citation provided.
  • OpenStax-copyrighted images, activities, assessments, and similar components of this book are subject to the same licensing – CC-BY-NC-SA. They can be used for noncommercial purposes with attribution. Commercial use requires permission.
  • Permission requests: Anyone who intends to incorporate this content (including text, images, and other components) into large language models, use it in AI offerings, use it commercially (including in print), and/or has questions about another use case is welcome to complete our reuse request form.
Attribution information
  • If you are redistributing all or part of this book in a noncommercial 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 noncommercial digital format, then for every page that includes OpenStax content, you must license the derivative work under the same CC-BY-NC-SA license as the original, and 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

The information below includes the information needed to generate citations in most major styles (APA, MLA, etc.); you must reformat and organize the information as needed to fit the requirements of the style. Use the information below to generate a citation. We recommend using a citation tool such as this one.

© Apr 23, 2026 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, and Rice University name, and Rice University logo trademarks, or wordmarks are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.