Multiply a Trinomial and a Binomial
Learning Outcome
- Multiply and divide polynomials
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(5x^{2}+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+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. [latex-display]\left(3x+6\right)\left(5x^{2}+3x+10\right)=15x^{3}+39x^{2}+48x+60[/latex-display]
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] The answer is [latex]15x^{3}+39x^{2}+48x+60[/latex].
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]
[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]
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
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
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.