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Polynomial Division:
    Simplification and Reduction
(page 1 of 3)

Sections: Simplification and reduction, Polynomial long division

There are two cases for dividing polynomials: either the "division" is really just a simplification and you're just reducting a fraction, or else you need to do long polynomial division (which is covered on the next page).

  • Simplify (2x + 4)/2

    This is just a simplification problem, because there is only one term in the polynomial that you're dividing by. And, in this case, there is a common factor in the numerator (top) and denominator (bottom), so it's easy to reduce this fraction. There are two ways of proceeding. I can split the division into two fractions, each with only one term on top, and then reduce:

      2x/2 + 4/2 = x + 2

    ...or else I can factor out the common factor from the top and bottom, and then cancel off:

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

    Either way, the answer is the same: x + 2

  • Simplify (21x^3 - 35x^2) / (7x)

    Again, I can solve this in either of two ways: by splitting up the sum and simplifying each fraction separately:   Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

      21x^3/7x - 25x^2/7x = 3x^2 - 5x

    ...or else by taking the common factor out front and canceling it off:

      7x(3x^2 - 5x)/7x = 3x^2 - 5x

    Either way, the answer is the same:  3x2 5x

Note: Most books don't talk about the domain at this point. But if your book does, you will need to note, for the above simplification, that x cannot equal zero. That is, for the simplified form to be completely mathematically equal to the original expression, the solution would need to be "3x2 5x, for all x not equal to 0".

  • Simplify  [ x(x + 3) - 2(x + 3) ] / (x + 3)

    I can split the sum and reduce each fraction separately:

      x(x + 3)/(x + 3) - 2(x + 3)/(x + 3) = x - 2

    The numerator (top) does indeed have a common factor; it's just a rather large one. Since both terms contain the factor "x + 3", then this is a common factor, and may be factored out front:

      (x + 3)(x - 2)/(x + 3) = x - 2

    Either way, the answer is the same: x 2

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Cite this article as:

Stapel, Elizabeth. "Polynomial Division: Simplification and Reduction." Purplemath. Available from Accessed


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