Multiplying Polynomials and Monomials

Lesson Plan - Get It!

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When you multiply polynomials (an expression that has more than one term), you have to pay close attention to what you are multiplying.

In this lesson, you will learn how to multiply two sets of monomials and a polynomial by a monomial.

You need to use exponent rules to multiply monomials (an expression with only one term, such as 3y²) and polynomials by monomials.

If you are not comfortable with exponents, please watch the video below as a review.

Multiplying Monomials

Multiplying monomials is similar to multiplying two numbers together but with a few added steps.

1. When you multiply variables, you must first identify and group the coefficients (plain old numbers).
2. Next, group like variables.
3. Finally, multiply all like sets.

Remember, a variable without an exponent carries an implied exponent of 1, so be sure to include the 1 when multiplying (which, in the case of exponents, is technically adding) like terms.

Try some examples.

Example: (5x²)(4x³)

1. Identify like terms (coefficients and xs): (5x²)(4x³)
2. Rearrange into coefficients and variables, in this case, xs: (5 * 4)(x² - x³)
3. Multiply the terms to get your answer: 20x⁵

Example with two different variables: (-3x³y²)(4xy⁵)

1. Identify like terms (coefficients, xs, and ys): (-3x³y²)(4xy⁵)
2. Rearrange: (-3 * 4)(x³ * x)(y² * y⁵)
3. Multiply terms to get the answer: -12x⁴y⁷

Try multiplying monomials on your own. Drag the correct answer into the answer space.

Multiplying Polynomials by Monomials

To multiply a polynomial by a monomial, use the distributive property and multiply each term inside the parenthesis by the term outside the parenthesis.

Before you begin, review the distributive property. This property of multiplication states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.

The distributive property says that if a, b, and c are real numbers, then a x (b + c) = (a x b) + (a x c).

If you still don't get it, watch the following video.

Once you're ready and feel comfortable with the distributive property, take a look at these examples.

Example 1: 2(3a² + 8a +11)

1. Distribute the outside monomial (2) to each term inside the parenthesis.

2. (2)3a²+ (2)8a + (2)11 — All like terms are together.

3. 6a²+ 16a + 22 — Multiply to get your final answer.

Example 2: 2x²y(3x-y)

1. 2x²y(3x) - 2x²y(y) — Distribute the outside monomial to each term inside the parenthesis.

2. (2 * 3)(x * x²)(y) - (2)(x²)(y * y) — Group like terms together.

3. 6x³y - 2x²y² — Multiply to get your final answer.

Example 3: 4a(a²b+2b²)

1. 4a(a²b) + 4a(2b²) — Distribute the outside monomial to each term inside the parenthesis.

2. (4)(a * a²)(b) + (4 * 2)(a)(b²) — Group like bases together.

3. 4a³b + 8ab² — Multiply together to get your final answer.

Try the next few practice problems. Drag the correct answer into the answer space.

Now that you understand how to multiply monomials — and polynomials by monomials —by grouping like bases, practice in the Got It? section.