Multiplication Division Monomials

Multiplication

Multiplication is a bit harder, because we have two operations going on, multiplication and addition. The polynomial is a sum of things, and we want to multiply two of these sums together, so we have to be a bit more careful about getting rid of the parentheses, because it involves changing the order of operation. With the parentheses the addition is being done first, but without them the normal order of operations rule says that the multiplication is being done first. With addition it was all one operation, so it didn't matter. What you end up doing in the end is multiplying all possible pairs, and in the multiplications you have to use the rules of exponents to simplify.

Division

We can also divide polynomials pretty much like we divide numbers. We do it like long division of several place numbers. It is nicer in some ways, but nastier in others. Remember how when you do long division, you make an estimate by using just the first places, and how it sometimes doesn't work, and you have to raise it or lower it? Well, with polynomials you do sort of the same thing, except it always works. But the nastier thing is that with polynomials you have to deal with negative number subtraction sometimes, which makes it easy to make careless errors. To makes things simpler I'll first do an example that doesn't involve any negative numbers.

Multiply monomials:

• Multiply the numerical coefficients.
• When variable factors are powers with the same base, multiply by adding exponent

Divide monomials:

• Divide their numerical coefficients.
• When variable factors are powers with the same base, divide by subtracting exponents.