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Polynomial Long Division (page 2 of 3)

Sections: Simplification and reduction, Polynomial long division

If you're dividing a polynomial by something more complicated than just a simple monomial, then you'll need a different method for simplification. The method is called "long division", and works just like the long division you did back in elementary school, except that now you're dividing with variables.

• Divide x² – 9x – 10 by x + 1

Think back to when you were doing long division with plain old numbers. You would be given one number that you had to divide into another number. You set up the division symbol and inserted the numbers where they belonged, and then started making guesses. And you didn't guess the whole answer right away; instead, you started working on the "front" part of the number you were dividing.   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Long division for polynomials works in much the same way:

Set up the division:
For the moment, ignore the other terms and look at the x and the x2. What do you have to multiply x by, to get x2? An x; put this on top:
Take that x, and multiply it through the x + 1.  First, you multiply x on the x; carry the x2 underneath:
Then multiply the x on the 1; carry the 1x underneath:
Draw the equals bar. To subtract, change all the signs in the second line...
...and add down. The first term will cancel out:
Remember to carry down that last term from the dividend:
Now look at the x from the divisor and the –10x from the bottom line in the division. What do you have to multiply x by to get –10x? A minus ten; put this on top:
Now multiply the –10 on the x, and carry the –10x to the bottom:
...and multiply the –10 on the 1, and carry the –10 to the bottom:
Draw the equals bar, and change the signs on all the terms in the bottom row:
Then add down:

Then the solution is: x – 10

Since the remainder is zero (that is, since there isn't anything left over), the division came out "even". When you do regular division with numbers and the division comes out even, it means that the number you divided by is a factor of the number you're dividing (for instance, if you divide 50 by 10 it will come out even, because 10 is a factor of 50). In this case, you now know that x + 1 is a factor of x² – 9x – 10, because the division came out even.

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