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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 to use a different method for the simplification. That method is called "long (polynomial) division", and it works just like the long (numerical) 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, inserted the two 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 (the larger place values) of the number you were dividing.   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Long division for polynomials works in much the same way:

First, I set up the division: For the moment, I'll ignore the other terms and look just at the leading x of the divisor and the leading x2 of the dividend.
If I divide the leading x2 inside by the leading x in front, what would I get? I'd get an x. So I'll put an x on top:
Now I'll take that x, and multiply it through the divisor, x + 1. First, I multiply the x (on top) by the x (on the "side"), and carry the x2 underneath:
Then I'll multiply the x (on top) by the 1 (on the "side"), and carry the 1x underneath:
Then I'll draw the "equals" bar, so I can do the subtraction. To subtract the polynomials, I change all the signs in the second line...
...and then I add down. The first term (the x2) will cancel out:
I need to remember to carry down that last term, the "subtract ten", from the dividend:
Now I look at the x from the divisor and the new leading term, the –10x, in the bottom line of the division. If I divide the –10x by the x, I would end up with a –10, so I'll put that on top:
Now I'll multiply the –10 (on top) by the leading x (on the "side"), and carry the –10x to the bottom:
...and I'll multiply the –10 (on top) by the 1 (on the "side"), and carry the –10 to the bottom:
I draw the equals bar, and change the signs on all the terms in the bottom row:
Then I add down:

Then the solution to this division is: x – 10

Since the remainder on this division was zero (that is, since there wasn'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, the answer will be a nice neat "5" with a zero remainder, because 10 is a factor of 50. In the case of the above polynomial division, the zero remainder tells us that x + 1 is a factor of x² – 9x – 10, which you can confirm by factoring the original quadratic dividend, x² – 9x – 10.

• Simplify 

This can be done in either of two ways: I can factor the quadratic and then cancel the common factor, like this:

But what if I didn't know how to factor? I can always use long division:

(I mustn't forget to change my signs, as shown in red, when I'm doing the subtraction.)

The answer to the division is quotient, the polynomial across the top:  x + 2

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