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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 2000-2011 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.

    Set up the division
    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:
    Put the 'x' up 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:
    Carry the 'x^2' down
    Then I'll multiply the x (on top) by the 1 (on the "side"), and carry the 1x underneath:
    Carry the '1x
    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...



    Change signs

    ...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:


    Carry down the '–10'
    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:  
    Put '–10' up top

    Now I'll multiply the 10 (on top) by the leading x (on the "side"), and carry the 10x to the bottom:


    Carry the '–10x' down

    ...and I'll multiply the 10 (on top) by the 1 (on the "side"), and carry the 10 to the bottom:


    Carry the '–10' down

    I draw the equals bar, and change the signs on all the terms in the bottom row:


    Change the signs

    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  (x^2 + 9x + 1)/(x + 7)

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

      (x + 2)(x + 7)/(x + 7) = x + 2

    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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Cite this article as:

Stapel, Elizabeth. "Polynomial Long Division." Purplemath. Available from Accessed


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