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Polynomial Division: Missing Dividends

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

At a middle-school level, this lesson will explain how to divide polynomials with missing terms in the dividends. Polynomials and monomials will be defined.

Polynomials

Suppose you are a florist and you need to make 8 centerpieces for a wedding. You have 32 red roses, 24 white roses and 16 pink roses. You want each centerpiece to have the same number of each color rose so that all the centerpieces match. How many of each color will be in each centerpiece?

In order to solve this problem, all we have to do is take each number and divide it by 8. This means you can make each centerpiece using 4 red roses, 3 white roses and 2 pink roses. We are going to use this same idea to divide polynomials.

A polynomial is an expression with numbers, variables or the product of numbers and variables which are added or subtracted. The variables in the polynomials must have whole number exponents.

Examples:


polynomials


A monomial is a polynomial with only one term.

Examples:


monomials


A polynomial can have any number of terms. Polynomials can also be written in standard form. Standard form of a polynomial is when the exponents of the variable are listed in order from largest to smallest.

The following polynomials are in standard form:


standard form


The following polynomials are not in standard form:


not standard


Polynomials do not have to have every exponent to be in standard form. Some polynomials will be missing terms. They might have an x cubed and an x, but no x squared term. We still put them in order from greatest exponent to least exponent and just leave out the missing terms.

Polynomial Division

In order to divide polynomials by monomials, we must divide each term of the polynomial by the monomial.

Example 1:

If we take the numbers from the floral example above, we can create a polynomial. If you think of the variables as the different colored roses, it might make sense that each color has a different exponent. So now, let's divide.


example1


For the above example, we are going to divide three times because there are three terms in the polynomial.

If we rewrite the problem in fraction form, it is easier to see what we are doing.


fraction


Now, we can rewrite it as separate fractions, because we are going to divide each term by 8x.


fractions


In order to divide, we simply divide the numbers. For the variables, when dividing, we subtract the exponents.


divison


Remember if you don't see an exponent, it's actually a 1. So, when we take x to the fourth power and divide it by x, we are actually subtracting one from 4 so we are left with x to the third power.

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