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).
When multiplying two of the same variables, add the exponents. Remember that the exponent on x is an invisible 1.
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.
Next we distribute the 4x.
Rewrite it again without the whole adding-a-negative thing to get our final answer: 24x – 8xy.
