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Study Guides > Intermediate Algebra

Read: Multiply a Trinomial and a Binomial

Learning Objectives

  • Multiply a binomial and a trinomial
Another type of polynomial multiplication problem is the product of a binomial and trinomial. Although the FOIL method can not be used since there are more than two terms in a trinomial, you still use the Distributive Property to organize the individual products. Using the distributive property, each term in the binomial must be multiplied by each of the terms in the trinomial. For our first examples, we will show you two ways to organize all of the terms that result from multiplying polynomials with more than two terms. The most important part of the process is finding a way to organize terms.

Example

Find the product.  [latex]\left(3x+6\right)\left(5x^{2}+3x+10\right)[/latex].

Answer: Distribute the trinomial to each term in the binomial. [latex-display]3x\left(5x^{2}+3x+10\right)+6\left(5x2+3x+10\right)[/latex-display] Use the distributive property to distribute the monomials to each term in the trinomials. [latex-display]3x\left(5x^{2}\right)+3x\left(3x\right)+3x\left(10\right)+6\left(5x^{2}\right)+6\left(3x\right)+6\left(10\right)[/latex-display] Multiply. [latex-display]15x^{3}9x^{2}+30x^{2}+18x+60[/latex-display] Group like terms. [latex-display]15x^{3}+\left(9x^{2}+30x^{2}\right)+\left(30x+18x\right)+60[/latex-display] Combine like terms.

Answer

[latex-display]\left(3x+6\right)\left(5x^{2}+3x+10\right)=15x^{3}+39x^{2}+48x+60[/latex-display]

As you can see, multiplying a binomial by a trinomial leads to a lot of individual terms! Using the same problem as above, we will show another way to organize all the terms produced by multiplying two polynomials with more than two terms.

Example

Multiply. [latex]\left(3x+6\right)\left(5x^{2}+3x+10\right)[/latex]

Answer: Set up the problem in a vertical form, and begin by multiplying [latex]3x+6[/latex] by [latex]+10[/latex]. Place the products underneath, as shown. [latex-display]\begin{array}{r}3x+\,\,\,6\,\\\underline{\times\,\,\,\,\,\,5x^{2}+\,\,3x+10}\\+30x+60\,\end{array}[/latex-display] Now multiply [latex]3x+6[/latex] by [latex]+3x[/latex]. Notice that [latex]\left(6\right)\left(3x\right)=18x[/latex]; since this term is like [latex]30x[/latex], place it directly beneath it. [latex-display]\begin{array}{r}3x\,\,\,\,\,\,+\,\,\,6\,\,\\\underline{\times\,\,\,\,\,\,5x^{2}\,\,\,\,\,\,+3x\,\,\,\,\,\,+10}\\+30x\,\,\,\,\,+60\,\,\\+9x^{2}\,\,\,+18x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex-display] Finally, multiply [latex]3x+6[/latex] by [latex]5x^{2}[/latex]. Notice that [latex]30x^{2}[/latex] is placed underneath [latex]9x^{2}[/latex]. [latex-display]\begin{array}{r}3x\,\,\,\,\,\,+\,\,\,6\,\,\\\underline{\times\,\,\,\,\,\,5x^{2}\,\,\,\,\,\,+3x\,\,\,\,\,\,+10}\\+30x\,\,\,\,\,+60\,\,\\+9x^{2}\,\,\,+18x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\underline{+15x^{3}+30x^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}[/latex-display] Now add like terms. [latex-display]\begin{array}{r}3x\,\,\,\,\,\,+\,\,\,6\,\,\\\underline{\times\,\,\,\,\,\,5x^{2}\,\,\,\,\,\,+3x\,\,\,\,\,\,+10}\\+30x\,\,\,\,\,+60\,\,\\+9x^{2}\,\,\,+18x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\underline{+15x^{3}\,\,\,\,\,\,+30x^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\+15x^{3}\,\,\,\,\,\,+39x^{2}\,\,\,\,+48x\,\,\,\,\,+60\end{array}[/latex-display]

Answer

[latex-display]15x^{3}+39x^{2}+48x+60[/latex-display]

Notice that although the two problems were solved using different strategies, the product is the same. Both the horizontal and vertical methods apply the Distributive Property to multiply a binomial by a trinomial. In our next example we will multiply a binomial and a trinomial that contains subtraction. Pay attention to the signs on the terms.  Forgetting a negative sign is the easiest mistake to make in this case.

Example

Find the product. [latex-display]\left(2x+1\right)\left(3{x}^{2}-x+4\right)[/latex-display]

Answer: [latex-display]\begin{array}{cc}2x\left(3{x}^{2}-x+4\right)+1\left(3{x}^{2}-x+4\right) \hfill & \text{Use the distributive property}.\hfill \\ \left(6{x}^{3}-2{x}^{2}+8x\right)+\left(3{x}^{2}-x+4\right)\hfill & \text{Multiply}.\hfill \\ 6{x}^{3}+\left(-2{x}^{2}+3{x}^{2}\right)+\left(8x-x\right)+4\hfill & \text{Combine like terms}.\hfill \\ 6{x}^{3}+{x}^{2}+7x+4 \hfill & \text{Simplify}.\hfill \end{array}[/latex-display]

Analysis of the Solution

Another way to keep track of all the terms involved in this product is to use a table, as shown below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify. Notice how we kept the sign on each term, for example we are subtracting [latex]x[/latex] from [latex]3x^2[/latex], so we place [latex]-x[/latex] in the table.
[latex]3{x}^{2}[/latex] [latex]-x[/latex] [latex]+4[/latex]
[latex]2x[/latex] [latex]6{x}^{3}[/latex] [latex]-2{x}^{2}[/latex] [latex]8x[/latex]
[latex]+1[/latex] [latex]3{x}^{2}[/latex] [latex]-x[/latex] [latex]4[/latex]
 

Example

Multiply.  [latex]\left(2p-1\right)\left(3p^{2}-3p+1\right)[/latex]

Answer: Distribute [latex]2p[/latex] and [latex]-1[/latex] to each term in the trinomial.

[latex]2p\left(3p^{2}-3p+1\right)-1\left(3p^{2}-3p+1\right)[/latex]

[latex]2p\left(3p^{2}\right)+2p\left(-3p\right)+2p\left(1\right)-1\left(3p^{2}\right)-1\left(-3p\right)-1\left(1\right)[/latex]

Multiply. (Notice that the subtracted [latex]1[/latex] and the subtracted [latex]3p[/latex] have a positive product that is added.)

[latex]6p^{3}-6p^{2}+2p-3p^{2}+3p-1[/latex]

Combine like terms.

[latex]6p^{3}-9p^{2}+5p-1[/latex]

Answer

[latex-display]6p^{3}-9p^{2}+5p-1[/latex-display]

In the following video we show more examples of multiplying polynomials. https://youtu.be/bBKbldmlbqI

Summary

Multiplication of binomials and polynomials requires use of the distributive property as well as the commutative and associative properties of multiplication. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in simplified form if all of its like terms have been combined.

Licenses & Attributions

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CC licensed content, Shared previously

  • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay Et al.. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Download for free at : http://cnx.org/contents/[email protected]:1/Preface.
  • Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology Located at: https://www.nroc.org/. License: CC BY: Attribution.
  • Ex: Polynomial Multiplication Involving Binomials and Trinomials. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.