Learning Outcome
- Multiply polynomials
Another type of polynomial multiplication problem is the product of a binomial and trinomial. Although the FOIL method cannot be used since there are more than two terms in a trinomial, you still use the Distributive Property and the Vertical Method 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. The most important part of the process is finding a way to organize terms.
example
Multiply using the Distributive Property: [latex]\left(x+3\right)\left(2{x}^{2}-5x+8\right)[/latex]
Solution
 |
|
| Distribute. | [latex]x\color{red}{(2x^2-5x+8)}+3\color{red}{(2x^2-5x+8)}[/latex] |
| Multiply. | [latex]2{x}^{3}-5{x}^{2}+8x+6{x}^{2}-15x+24[/latex] |
| Combine like terms. | [latex]2{x}^{3}+{x}^{2}-7x+24[/latex] |
Example
Find the product. [latex]\left(3x+6\right)\left(5x^{2}+3x+10\right)[/latex]
try it
As you can see, multiplying a binomial by a trinomial leads to a lot of individual terms! Using the same problems as above, we will use the vertical method to organize all the terms produced by multiplying two polynomials with more than two terms.
example
Multiply using the Vertical Method: [latex]\left(x+3\right)\left(2{x}^{2}-5x+8\right)[/latex]
Example
Multiply. [latex]\left(3x+6\right)\left(5x^{2}+3x+10\right)[/latex]
try it
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]\left(2x+1\right)\left(3{x}^{2}-x+4\right)[/latex]
Another way to keep track of all the terms involved in the above 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]
In the following video, we show more examples of multiplying polynomials.
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.
