Polynomials can sometimes be separated using the simple methods displayed on dividing Polynomials.

But sometimes it is much better to use "Long Division" (a technique similar to Long department for Numbers)

Numerator and also Denominator

We can offer each polynomial a name:

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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^3 divided by x

The Method

Write it under neatly:

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

Both polynomials should have the "higher order" terms an initial (those through the biggest exponents, prefer the "2" in x2).

Then:


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Divide the first term of the molecule by the an initial term of the denominator, and also put the in the answer.Multiply the denominator by the answer, placed that listed below the numeratorSubtract to develop a new polynomial
Repeat, utilizing the new polynomial

It is less complicated to show with an example!

Example:

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

Check the answer:

Multiply the prize by the bottom polynomial, we should acquire the peak polynomial:

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Remainders

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

After splitting we were left through "2", this is the "remainder".

The remainder is what is left over after dividing.

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

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"Missing" Terms

There deserve to be "missing terms" (example: there may be one x3, however no x2). In that instance either leaving gaps, or incorporate the absent terms v a coefficient the zero.

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Example:

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Write that down with "0" coefficients for the absent terms, then resolve it usually (press play):

See exactly how we needed a an are for "3x3"?

More than One Variable

So much we have been splitting polynomials with just one variable (x), yet we can handle polynomials v two or much more variables (such as x and also y) utilizing the exact same method.