*Contributor: Mason Smith. Lesson ID: 11549*

Working with polynomials may seem hard, so why multiply them and make more? That's not quite what we mean, but you will learn how to multiply the expressions by first working with monomials and games!

categories

subject

Math

learning style

Visual

personality style

Otter

Grade Level

Middle School (6-8), High School (9-12)

Lesson Type

Quick Query

Kristen: What is *u* times *r* times *r*?

Alan: *ur*^{2}

Kristen: No, I'm not!

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 together, and how to multiply a polynomial by a monomial. If you have not completed the previous lessons in this *Polynomials* series, please go to **Related Lessons** in the right-hand sidebar.

To multiply *monomials* (an expression with only one term, such as 3y^{2}) and polynomials by monomials, you need to use exponent rules. If you are not comfortable with exponents, please watch* Exponent Rules Song – Learn Algebra – Learning Upgrade* as a review:

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

- When you multiply variables, you must first identify the
*coefficients*(plain old numbers), and group them together. - Next, group like
*variables*. - 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.

Let's try some examples:

Example: (5x^{2})(4x^{3})

- Identify like terms (coefficients and x's:) (5x
^{2})(4x^{3}) - Rearrange into coefficients and variables, in this case, x's: (5 * 4)(x
^{2}- x^{3}) - Multiply the terms to get your answer: 20x
^{5}

Example with two different variables: (-3x^{3}y^{2})(4xy^{5})

- Identify like terms (coefficients, x's, and y's): (-3x
^{3}y^{2})(4xy^{5}) - Rearrange: (-3 * 4)(x
^{3}* x)(y^{2}* y^{5}) - Multiply terms to get answer: -12x
^{4}y^{7}

Try your hand at multiplying monomials and share your results with a parent or teacher. Drag the correct answer into the answer space:

To multiply a polynomial by a monomial, use the *d**istributive **p**roperty* and multiply each term inside the parenthesis by the term outside the parenthesis.

Before we begin, let's review the Distributive Property. It is the property of multiplication that states that multiplying a sum by a number is the same as multiplying each addend by the number, 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, here's a wacky video all about the *Distributive Property of Multiplication* that should clear things up nicely.

Once you're ready, and feel comfortable with the Distributive Property, take a look at these examples:

Example 1: 2(3a^{2} + 8a +11)

Distribute the outside monomial (2) to each term inside the parenthesis.- (2)3a
^{2}+ (2)8a + (2)11 — All like terms are together. - 6a
^{2}+ 16a + 22 — Multiply to get your final answer.

Example 2: 2x^{2}y(3x-y)

- 2x
^{2}y(3x) - 2x^{2}y(y) — Distribute the outside monomial to each term inside the parenthesis. - (2 * 3)(x * x
^{2})(y) - (2)(x^{2})(y * y)

Group like terms together. - 6x
^{3}y - 2x^{2}y^{2}— Multiply to get your final answer.

Example 3: 4a(a^{2}b+2b^{2})

- 4a(a
^{2}b) + 4a(2b^{2}) — Distribute the outside monomial to each term inside the parenthesis. - (4)(a * a
^{2})(b) + (4 * 2)(a)(b^{2}) — Group like bases together. - 4a
^{3}b + 8ab^{2}— Multiply together to get your final answer.

Try the next few practice problems and share your answer with a parent or teacher. Drag the correct answer into the answer space:

Now that you have an understanding of how to multiply monomials — and polynomials by monomials —by grouping like bases, let's practice.

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