SOLUTION: a^2-7a-18/4a^3 divided by a^2-4a-45/2a^3-4a^2 times a^2+3a-10/a^2-4a+4

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: a^2-7a-18/4a^3 divided by a^2-4a-45/2a^3-4a^2 times a^2+3a-10/a^2-4a+4      Log On


   

Question 624262: a^2-7a-18/4a^3 divided by a^2-4a-45/2a^3-4a^2 times a^2+3a-10/a^2-4a+4
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!

To divide fractions we learn to change it to multiplication of the reciprocal. So this becomes:


Back in the "good old days", before we had variables in the fractions, we learned that, when you are multiplying fractions,
  • it was OK to cancel before multiplying the fractions.
  • it was OK to "cross-cancel" (i.e. cancel a factor of one fraction's numerator with a factor of another fraction's denominator. For example:
    %281%2F2%29%2A%283%2F5%29%2A%282%2F7%29%2A%285%2F11%29

    %281%2F1%29%2A%283%2F1%29%2A%281%2F7%29%2A%281%2F11%29
    3%2F77
  • that this "pre-multiplication" canceling made the multiplication easier and the post-multiplication fraction easier to reduce.
Well not only is this still true with more complicated fractions, the advantages of "pre-multiplication" canceling are more important than ever!! So before we even think about multiplying these fractions together we want to cancel as many factors as we can. And to cancel factors we have to know what they are. So the next step is to factor each numerator and denominator as much as possible:
a%5E2-7a-18 factored as a trinomial: (a-9)(a+2)
4a%5E3 is already factored although we could factor the 4 and "factor out the exponent": 2%2A2%2Aa%2Aa%2Aa
2a%5E3-4a%5E2 has a common factor of 2a%5E2: 2a%5E2%28a-2%29 or 2*a*a(a-2)
a%5E2-4a-45 factored as a trinomial: (a-9)(a+5)
a%5E2%2B3a-10 factored as a trinomial: (a+5)(a-2)
a%5E2-4a%2B4 factored as a trinomial (or with the a%5E2-1ab_b%5E2+=+%28a-b%29%28a-b%29 pattern): (a-2)(a-2)
Rewriting

with factored numerators and denominators:

We should see that we have quite a few factors to cancel out!

leaving:
%28%28a-9%29%2F%282a%29%29%2A%281%2F1%29%2A%281%2F1%29
which, you must agree, is much easier to multiply than
.
Multiplying out reduced fractions we get:
%28a-9%29%2F%282a%29
If we fully factored and canceled, then there should be no need to reduce after multiplying. And as you can see, there are no factors that will cancel here. (The "a" in the numerator is not a factor so you cannot cancel it with the "a" factor in the denominator.)