SOLUTION: multiply or divide as indicated y^2-16/y^3-64 ÷ y^2-3y-18/y^2+5y+6

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Question 524782: multiply or divide as indicated
y^2-16/y^3-64 ÷ y^2-3y-18/y^2+5y+6

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Although you did not write it the following way, I think the problem below is the one you intended to get help with. If I am incorrect, please post your problem again.
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+%28%28y%5E2-16%29%2F%28y%5E3-64%29%29%2F%28%28+y%5E2-3y-18%29%2F%28y%5E2%2B5y%2B6%29%29+
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This is a division problem involving two fractions, one in the numerator and one in the denominator. The dividend (the number being divided) is the numerator and the divisor (the number that is to be divided into the dividend) is the denominator. For this problem we have:
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The dividend (or numerator) is:
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%28%28y%5E2-16%29%2F%28y%5E3-64%29%29
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and the divisor (or denominator) is:
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%28%28+y%5E2-3y-18%29%2F%28y%5E2%2B5y%2B6%29%29+
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Whenever you divide by a fraction (that is when the divisor is a fraction), the rule is that you invert the fraction that is the divisor and multiply it times the dividend. Inverting the divisor (flipping it upside down) in this problem results in changing it to:
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%28%28y%5E2%2B5y%2B6%29%2F%28y%5E2-3y-18%29%29+
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Now multiply this times the dividend in this problem and you have converted your problem from a division of two fractions to a multiplication of two fractions as follows:
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%28%28y%5E2-16%29%2F%28y%5E3-64%29%29%2A%28%28y%5E2%2B5y%2B6%29%2F%28y%5E2-3y-18%29%29+
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The reason for this conversion is just that it makes it a little easier to work with rather than have the "tall multi-level" problem we were given originally.
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The next thing to do is to start factoring the terms in the numerators and denominators of these two fractions to see if we can cancel out some of them to make the problem simpler.
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First factor y%5E2+-+16. Note that this is the difference of two squares and the rule is y%5E2+-+c%5E2+=+%28y+-+c%29%28y%2Bc%29. In this case c is 4 so we can say that y%5E2+-+16+=%28y-4%29%28y%2B4%29. Substitute this and the problem becomes:
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You can work on verifying the factoring of the other terms in this problem. Here is what they are:
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y%5E3+-+64+=+%28y-4%29%28y%5E2%2B4y%2B16%29
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y%5E2+%2B5y%2B6%29+=+%28y%2B3%29%28y%2B2%29
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and
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y%5E2+-3y-18+=+%28y-6%29%28y%2B3%29
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Substitute these factorized results into the problem and it becomes:
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Now we can cross out any factors in the numerators that are the same as factors in the denominators as shown below:
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and we are left with:
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+%28%28y%2B4%29%2F%28y%5E2%2B4y%2B16%29%29%2A%28%28y%2B2%29%2F%28y-6%29%29+
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which could also be written as:
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+%28%28y%2B4%29%28y%2B2%29%29%2F%28%28y-6%29%28y%5E2%2B4y%2B16%29%29+
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This is the answer to your problem.
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Hope this helps you to understand what the problem was asking you to do and a process that would help you get to the answer.