Polynomials - Long Division

A polynomial looks like this:

polynomial example
example of a polynomial
this one has 3 terms

Dividing

polynomial-long-division

Polynomials can sometimes be divided using the simple methods shown on Dividing Polynomials.

But sometimes it is best to use "Long Division" (a method similar to Long Division for Numbers)

Numerator and Denominator

We can give each polynomial a name:

polynomial-long-division
  • the top polynomial is the numerator
  • the bottom polynomial is the denominator

If you have trouble remembering, think denominator is down-ominator.

The Method

Write it down neatly:

polynomial-long-division
  • the denominator goes first,
  • then a ")",
  • then the numerator with a line above

Both polynomials should have the "higher order" terms first (those with the largest exponents, like the "2" in x2).

Then:

repeat
  • Divide the first term of the numerator by the first term of the denominator, and put that in the answer.
  • Multiply the denominator by that answer, put that below the numerator
  • Subtract to create a new polynomial
  Repeat, using the new polynomial

It is easier to show with an example!

Example:

poly long division

Write it down neatly like below, then solve it step-by-step (press play):

Check your answer:

Just multiply your answer by the bottom polynomial, you should get the top polynomial:

poly long division yes

Remainders

The previous example worked perfectly, but that is not always so! Try this one:

After dividing we were left with "2", this is the "remainder".

The remainder is what is left over after dividing.

But you still have an answer, just put the remainder divided by the bottom polynomial as part of the answer, like this:

poly long div b

"Missing" Terms

Sometimes there will be "missing terms" (example: there may be an x3, but no x2). In that case either leave gaps, or include the missing terms with a coefficient of zero.

Example:

poly long div c

Write it down with "0" coefficients for the missing terms, then solve it normally (press play):

When solving it I just left gaps, and as you can see it was important to have the space available because we needed it for "3x3"

More than One Variable

So far we have been dividing polynomials with only one variable (x), but how do you handle polynomials with two or more variables (such as x and y)? You can still use the same method.

Example:

poly long div d