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Math Help - Polynomial division and dividing by zero

  1. #1
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    Polynomial division and dividing by zero

    Suppose I define  g(x,y) = \frac {f(x)-f(y)}{x-y} and take  f(z) = z^{n} . So  g(x,y) = x^{n-1}+ x^{n-2}y + \cdots + xy^{n-2} + x^{k-1} . If I take  y=x , then  g(x,x) = nx^{n-1} . Is this valid? If I look at the original equation, does this also mean  g(x,x) = \frac {0}{0} ? I'm not sure how to understand division by zero.
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  2. #2
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    g(x,x) \neq nx^{n-1} because deriving g(x,y) = x^{n-1}+ x^{n-2}y + \cdots + xy^{n-2} + x^{k-1} from g(x,y) = \frac {f(x)-f(y)}{x-y} is valid only when x \neq y.

    Again, if x=y then g is not defined, therefore you can't say that g(x,x) = nx^{n-1}
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  3. #3
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    OK, the problem I'm doing says  g(x,y) is always divisible by  x - y , so instead, can I say because  (x-y)g(x,x) = (x-y)(nx^{n-1}) , and then using the cancellation law, g(x,x) is the same polynomial as nx^{n-1}?
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