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Dividing Rational Expressions (page 2 of 2)

For dividing rational expressions, you will use the same method as you used for dividing numerical fractions: when dividing by a fraction, you flip-n-multiply. For instance:

  • Perform the indicated operation:

      • (4/3) ÷ (9/5)

    To simplify this division, I'll convert it to multiplication by flipping what I'm dividing by; that is, I'll switch from dividing by a fraction to multiplying by that fraction's reciprocal. Then I'll simplify as usual:

      (4/3) ÷ (9/5) = (4/3) × (5/9) = 20/27

Can the 2's cancel off from the 20's? No! This is as simplified as the fraction gets.

Division works the same way with rational expressions.

  • Perform the indicated operation:

      • [ (x^2 + 2x - 15) / (x^2 - 4x - 45) ] ÷ [ (x^2 + x - 12) / (x^2 - 5x - 36) ]

    To simplify this, first I'll flip-n-multiply. Then, to simplify the multiplication, I'll factor the numerators and denominators, and then cancel any duplicated factors. My work looks like this:

      [(x^2+2x-15)/(x^2-4x-45)] ÷ [(x^2+x-12)/(x^2-5x-36)] = [(x^2+2x-15)/(x^2-4x-45)] × [(x^2-5x-36)/(x^2+x-12)] = [(x+5)(x-3)(x-9)(x+4)]/[(x-9)(x+5)(x+4)(x-3)] = 1

    Then the answer is: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

      1, for x not equal to -5, -4, 3, or 9

Your instructor may not require the restrictions on the allowable x-values, in which case your answer would be just "1". The exercises you'll be given won't usually simplify this much, of course. The following example is much more typical:

  • Simplify the following expression:

      • [ (x^2 + 3x - 40) / (x^2 + 2x - 35) ] ÷ [ (x^2 + 2x - 48) / (x^2 + 3x - 18) ]

    First, I'll need to flip the second fraction, and convert from division to multiplication. Then I'll factor, and see if anything cancels.

      [(x^2+3x-40)/(x^2+2x-35)] ÷ [(x^2+2x-48)/(x^2+3x-18)] = [(x^2+3x-40)/(x^2+2x-35)] × [(x^2+3x-18)/(x^2+2x-48)] = [(x+8)(x-5)(x+6)(x-3)/(x+7)(x-5)(x+8)(x-6)] = [(x+6)(x-3)]/[(x+7)(x-6)]

(Can you cancel off the 6's? or the x's? No! The above is as simplified as this gets!)

    Then the final answer is:

      [ (x + 6)(x - 3) ] / [ (x + 7)(x - 6) ] = [x^2 + 3x - 18] / [ (x + 7)(x - 6) ] for x not equal to -8, -6, 3, or 5

For reasons which will become clear when adding and subtracting rationals, the numerator is usually multiplied out ("simplified" to get rid of the parentheses), while the denominator is usually left in factored form.

Make sure you know how to factor quadratics and cubics, because, as you have seen, it is required for many of the problems you'll be doing. Also, make sure you are careful to cancel only factors, not terms. If you can keep that straight, then you'll probably do fine.

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Cite this article as:

Stapel, Elizabeth. "Dividing Rational Expressions." Purplemath. Available from
    http://www.purplemath.com/modules/rtnlmult2.htm. Accessed
 

 

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