6.3 Multiply Polynomials

Learning Objectives

By the end of this section, you will be able to:

  • Multiply a polynomial by a monomial
  • Multiply a binomial by a binomial
  • Multiply a trinomial by a binomial

Multiply a Polynomial by a Monomial

We have used the Distributive Property to simplify expressions like 2(x−3). You multiplied both terms in the parentheses, xand3, by 2, to get 2x−6. With this chapter’s new vocabulary, you can say you were multiplying a binomial, x−3, by a monomial, 2.

Multiplying a binomial by a monomial is nothing new for you! Here’s an example:

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.

Multiply a Binomial by a Binomial Using the Distributive Property

Look at Example 6.33, where we multiplied a binomial by a monomial.

Multiply a Binomial by a Binomial Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes, like in Example 6.37, there are no like terms to combine.

Let’s look at the last example again and pay particular attention to how we got the four terms.

Where did the first term, x2, come from?

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products.

Let’s look at (x+3)(x+7).

Notice how the terms in third line fit the FOIL pattern.

Now we will do an example where we use the FOIL pattern to multiply two binomials.

We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

How To

Multiply two binomials using the FOIL method

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When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.

The final products in the last four examples were trinomials because we could combine the two middle terms. This is not always the case.

Be careful of the exponents in the next example.

Multiply a Binomial by a Binomial Using the Vertical Method

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 polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.This figure shows the vertical multiplication of 23 and 46. The number 23 is above the number 46. Below this, there is the partial product 138 over the partial product 92. The final product is at the bottom and is 1058. Text on the right side of the image says “Start by multiplying 23 by 6 to get 138. Next, multiply 23 by 4, lining up the partial product in the correct columns. Last you add the partial products.”

Now we’ll apply this same method to multiply two binomials.

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.

Multiplying Two Binomials

To multiply binomials, use the:

  • Distributive Property
  • FOIL Method
  • Vertical Method

Remember, FOIL only works when multiplying two binomials.

Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a trinomial by a binomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

Now let’s do this same multiplication using the Vertical Method.

We have now seen two methods you can use to multiply a trinomial by a binomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.

Multiplying a Trinomial by a Binomial

To multiply a trinomial by a binomial, use the:

  • Distributive Property
  • Vertical Method

Media

Access these online resources for additional instruction and practice with multiplying polynomials:

  • Multiplying Exponents 1
  • Multiplying Exponents 2
  • Multiplying Exponents 3

Section 6.3 Exercises

Practice Makes Perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

In the following exercises, multiply the binomials. Use any method.

Multiply a Trinomial by a Binomial

In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

In the following exercises, multiply. Use either method.

Mixed Practice

Everyday Math

Writing Exercises

296.Which method do you prefer to use when multiplying two binomials: the Distributive Property, the FOIL method, or the Vertical Method? Why?
297.Which method do you prefer to use when multiplying a trinomial by a binomial: the Distributive Property or the Vertical Method? Why?
298.Multiply the following:

Explain the pattern that you see in your answers.
299.Multiply the following:

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “multiply a polynomial by a monomial,” “multiply a binomial by a binomial,” and “multiply a trinomial by a binomial.” The rest of the cells are blank.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?