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Adding and Subtracting
     Rational Expressions: More Examples (page 3 of 3)

• Simplify the following:

To find the common denominator, first I'll have to factor the quadratic denominator:

x² – 5x – 6 = (x – 6)(x + 1)

Fortunately for me, the quadratic denominator didn't introduce any new factors to the problem, so the common denominator will be (x – 6)(x + 1).

Since I was able to cancel out the x + 1 factor, I eliminated a zero from the denominator. Then the final answer is: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

You might not need the "for x not equal to –1" part of the solution.

• Simplify the following:

First I'll factor the quadratic denominator:

x² + 3x – 10 = (x + 5)(x – 2)

Note that these factors almost match the other denominators, but the second fraction's denominator is backwards. How can I fix that? By remembering the following:

5 – 3 = 2
3 – 5 = –2

The point of this is that, when I reversed the subtraction, I got the same answer except for the sign. So I can reverse the subtraction in the second fraction's denominator, as long as I remember to reverse the sign. This is what that looks like:

I factored the numerator, but nothing cancels out. As you can see, I had to factor a denominator, multiply two of the fractions to get a common denominator, multiply those two fractions' numerators, add, simplify, and then factor again. You should expect to see some problems that are at least this involved. They're not as much complicated as they are long and annoying. Work them out step-by-step as I did above, and you'll get the right answers fairly regularly. In this case, the answer is:

When you're adding and subtracting rationals, don't try to do a lot of steps in your head, or skip steps or do half-steps (like leaving out the denominators in your calculations), or you'll pretty much guarantee yourself the wrong answer. Take the time to do every step, and practice them on the homework, so you have a good chance of getting these right on the test.

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