Polynomials can sometimes be split using the an easy methods shown on dividing Polynomials.

But periodically it is much better to use "Long Division" (a method similar come Long department for Numbers)

Numerator and Denominator

We can provide each polynomial a name:


the top polynomial is the numeratorthe bottom polynomial is the denominator

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

You are watching: X^2 divided by x^2

The Method

Write it down neatly:


the denominator goes first, climate a ")", then the numerator through a heat above

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


Divide the very first term of the molecule by the very first term the the denominator, and put that in the answer.Multiply the denominator by that answer, placed that listed below the numeratorSubtract to produce a brand-new polynomial
Repeat, using the brand-new polynomial

It is simpler to show with an example!



Write that down nicely like below, then fix it step-by-step (press play):

Check the answer:

Multiply the answer by the bottom polynomial, us should obtain the optimal polynomial:



The previous instance worked perfectly, but that is not always so! shot this one:

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

The remainder is what is left end after dividing.

But us still have an answer: placed the remainder separated by the bottom polynomial as component of the answer, prefer this:


"Missing" Terms

There deserve to be "missing terms" (example: there might be an x3, yet no x2). In that situation either leave gaps, or incorporate the lacking terms v a coefficient that zero.

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Write that down through "0" coefficients because that the absent terms, then resolve it generally (press play):

See how we required a an are for "3x3"?

More 보다 One Variable

So far we have actually been dividing polynomials with just one change (x), however we have the right to handle polynomials through two or an ext variables (such together x and also y) making use of the same method.