Polynomials and Polynomial Functions
Multiply Polynomials
Learning Objectives
By the end of this section, you will be able to:
- Multiply monomials
- Multiply a polynomial by a monomial
- Multiply a binomial by a binomial
- Multiply a polynomial by a polynomial
- Multiply special products
- Multiply polynomial functions
Before you get started, take this readiness quiz.
Multiply Monomials
We are ready to perform operations on polynomials. Since monomials are algebraic expressions, we can use the properties of exponents to multiply monomials.
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Multiply a Polynomial by a Monomial
Multiplying a polynomial by a monomial is really just applying the Distributive Property.
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| Distribute. | |
| Multiply. |
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Multiply a Binomial by a Binomial
Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.
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| Distribute | |
| Distribute again. | |
| Combine like terms. |
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| Distribute. | |
| Distribute again. | |
| Combine like terms. |
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If you multiply binomials often enough you may notice a pattern. Notice that the first term in the result is the product of the first terms in each binomial. The second and third terms are the product of multiplying the two outer terms and then the two inner terms. And the last term results from multiplying the two last terms,
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. We use this as another method of multiplying binomials. The word FOIL is easy to remember and ensures we find all four products.
Let’s multiply using both methods.
We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!
When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.
Now we will do an example where we use the FOIL pattern to multiply two binomials.
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The final products in the last example were trinomials because we could combine the two middle terms. This is not always the case.
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| Step 1. Multiply the First terms. | |
| Step 2. Multiply the Outer terms. | |
| Step 3. Multiply the Inner terms. | |
| Step 4. Multiply the Last terms. | |
| Step 5. Combine like terms—there are none. |
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| Step 1. Multiply the First terms. | |
| Step 2. Multiply the Outer terms. | |
| Step 3. Multiply the Inner terms. | |
| Step 4. Multiply the Last terms. | |
| Step 5. Combine like terms. |
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The FOIL method is usually the quickest method for multiplying two binomials, but it only works for binomials. You can use the Distributive Property to find the product of any two polynomia