Study Guide

Basic Algebra - Multiplying Monomials

Multiplying Monomials

A polynomial is an expression that's made up of constants and/or variables. All the expressions we've been dealing with so far have been polynomials: 5x + 17 and 18xy2 – 17xy + 19y are both polynomials, for example. And we saw from our handy chart earlier that a monomial is an expression that's made of a single term, like 5x. ("Mono-" just means "one.")

When we learned about the distributive property, we were multiplying polynomials, but now we'll look at this a bit deeper. Look at the examples carefully and make note of the exponents. Remember: 5xy means 5 times x times y.

Again, it's helpful to think of subtraction as adding a negative: (x – 5) is the same as x + (-5). This will help us keep track of which terms are negative and which are positive.

Multiplying a Monomial by a Monomial

When multiplying a monomial by a monomial, we multiply the coefficients together and tack on the variables at the end (usually in alphabetical order).

(14a)(2b) = 28ab

When multiplying two of the same variables, add the exponents. Remember that the exponent on x is an invisible 1.

x x^2 = x^1 x^2 = x^3

The reason for this is that x2 is really just x times x, and x times x2 is x times x times x, or xxx, which equals x3 (since there are three x's). The exponent tells us how many variables to multiply together.

Multiplying a Monomial by a Polynomial

This is the same thing as the distributive property that we just learned. Let's say we want to multiply 4x(6 – 2y). First we're going to change the subtraction symbol to adding a negative.

4x(6 + -2y)

Next we distribute the 4x.

distribution arrows 4x(6 + -2y)

Rewrite it again without the whole adding-a-negative thing to get our final answer: 24x – 8xy.

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