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Complex Fractions: Technicalities

I started with this:

  • Simplify the following expression:

      • [ 1/(x - 1) + x + 3 ] / [ x - 3 + 1/(x + 4) ]

...and ended up with this:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

       (x^3 + 6x^2 + 6x - 8) / (x^3 - 12x + 11) for x not equal to 1, -4, or (-1 ± 3sqrt(5)) / 2

The first two domain restrictions were easy. The sub-fraction in the complex fraction's numerator will be undefined if x – 1 = 0, and the sub-fraction in the denominator will be undefined if x + 4 = 0. The other restriction is harder. I had to consider the entire denominator. This would be zero when the numerator (of the denominator) was zero. Converting to a fraction, I got this:

    [(x^2 + 2x - 2)/(x - 1)] / [(x^2 + x -11)/(x + 4)]

The denominator will be zero when x2 + x – 11 is zero. Solving, I get:

    x = [-1 ± 3sqrt(5)] / 2

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