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Math Help - polynomial and rational zeros

  1. #1
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    Aug 2009
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    polynomial and rational zeros

    1. express 3/23 as the sum of two unit fractions, 3/23=1/a+1/b

    2. consider y=x^2. Determine the poin on the function closest to (3,1)

    I know answers to the question 1 (
    8 and 184) but I don't know how to solve it.

    Can someone teach me how to solve them?
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  2. #2
    Senior Member
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    Apr 2009
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    Atlanta, GA
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    409
    \frac{3}{23}=\frac1a+\frac1b=\frac{a+b}{ab}, so a+b=3k, ab=23k for some integer k that will cancel on top and bottom. Substitute b=\frac{23k}{a}, so a+\frac{23k}{a}=3, giving the quadratic a^2-3ka+23k=0. The determinant is D=9k^2-92k, so the first value for k that makes D a perfect square will give you your answer!

    Consider the distance D between the points (x,x^2),(3,1). Let D^2=(x-3)^2+(x^2-1)^2. To minimize D^2, find its derivative and set to zero. You won't get an integer value, and need a calculator.

    Hope this helps!
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