Learning Outcome
- Multiply polynomials
Multiplying Polynomials
In the last two sections we learned how to use the Distributive Property to multiply monomials and binomials. The Distributive Property can be expanded to find the product of any two polynomials; each term in the first polynomial must be multiplied into each term in the second polynomial.
Example
Find the product: [latex](3x+4)(2x^2-3x-8)[/latex]
Solution
Distributive Property:
[latex]\begin{equation}\begin{aligned}&\;\;\;\;(\color{red}{3x}\color{blue}{+4})(2x^2-3x-8)\\&=\color{red}{3x}(2x^2)+\color{red}{3x}(-3x)+\color{red}{3x}(-8)\color{blue}{+4}(2x^2)\color{blue}{+4}(-3x)\color{blue}{+4}(-8)\\&=6x^3-9x^2-24x+8x^2-12x-32\\&=6x^3-x^2-36x-32\end{aligned}\end{equation}[/latex]
Answer
[latex](3x+4)(2x^2-3x-8)=6x^3-x^2-36x-32[/latex]
Example
Find the product: [latex](5x^2-x+3)(2x^2+4x-7)[/latex]
Solution
Distributive Property:
[latex]\begin{equation}\begin{aligned}&\;\;\;\;(\color{red}{5x^2}\color{blue}{-x}+\color{green}{3})(2x^2+4x-7)\\&=\color{red}{5x^2}(2x^2)+\color{red}{5x^2}(4x)+\color{red}{5x^2}(-7)\color{blue}{-x}(2x^2)\color{blue}{-x}(4x)\color{blue}{-x}(-7)+\color{green}{3}(2x^2)+\color{green}{3}(4x)+\color{green}{3}(-7)\\&=10x^4+20x^3-35x^2-2x^3-4x^2+7x+6x^2+12x-21\\&=10x^4+20x^3-2x^3-35x^2-4x^2+6x^2+7x+12x-21\\&=10x^4+18x^3-33x^2+19x-21\end{aligned}\end{equation}[/latex]
Answer
[latex](5x^2-x+3)(2x^2+4x-7)=10x^4+18x^3-33x^2+19x-21[/latex]
Try It
Find the product: [latex](x+5)(x^2-3x+4)[/latex]
Try It
Find the product: [latex](2x^2-3x+1)(x^2-4x+3)[/latex]
try it
The vertical method can also be extended to any size polynomial. The next example shows the multiplication of a trinomial and a binomial.
example
Multiply using the Vertical Method: [latex]\left(x+3\right)\left(2{x}^{2}-5x+8\right)[/latex]
Solution
It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.
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| Multiply [latex]\left(2{x}^{2}-5x+8\right)[/latex] by [latex]3[/latex]. | [latex]6x^2-15x+24[/latex] |
| Multiply [latex]\left(2{x}^{2}-5x+8\right)[/latex] by [latex]x[/latex] . | [latex]2x^3-5x^2+\;8x\;\;\;\;\;\;\;\;[/latex] |
| Add like terms. | [latex]2x^3\;+\;\;x^2-\;7x+24[/latex] |
try it
Watching signs and keeping track of all the terms require organization and attention to detail.
Example
Find the product.
[latex]\left(2x+1\right)\left(3{x}^{2}-x+4\right)[/latex]
Solution
[latex]\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]
Another way to keep track of all the terms involved in the above product is to use a table. 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]
Solution
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]
Try It
Multiply. [latex]\left(2y-5\right)\left(y^{2}-2y+3\right)[/latex]
The following video shows more examples of multiplying polynomials.
Try It
Multiply: [latex]2x(x+3)(x-3)[/latex]
Try It
Multiply: [latex]x^3(x+4)(x^2-3x+1))[/latex]
Summary
Multiplication of binomials and polynomials requires an understanding of the distributive property, rules for exponents, and a keen eye for collecting like terms. 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 standard form when all of its like terms have been combined and the resulting terms are written in descending order.
