To reduce a fraction, you eliminate factors common to the top and bottom. So
the first step in reducing an algebraic fraction is to factor the numerator and
denominator. Here the denominator is easy since it’s the difference of
squares: 9 – x^2 = (3 – x)(3 + x). The numerator takes some thought and some
trial and error. For the first term to be 3x^2 , the first terms of the factors must
be 3x and x. For the last term to be +6, the last terms must be either +2 and +3,
or –2 and –3, or +1 and +6, or –1 and –6. After a few tries, you should come up
with: 3 x^2 – 11x + 6 = (3x – 2)(x – 3). Now the fraction looks like this:
In this form there are no precisely common factors, but there is a factor in the
numerator that’s the opposite (negative) of a factor in the denominator: x – 3
and 3 – x are opposites. Factor –1 out of the numerator and get: