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Math Help - Rational zero test and finding solutions

  1. #1
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    Rational zero test and finding solutions

    I have two problems on my review that I just cannot seem to remember how to work...If anyone could please help, I would really appreciate it!
    1. Use the Rational Zero Test to determine all possible rational zeros of the function and then find all the actual zeros of f(x) = x^3 + 2x^2 - 21x + 18
    2. Find all the solutions to f(x) = -8x^4 + 128x^2
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  2. #2
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    Hello, aikenfan!

    1. Use the Rational Zero Test to determine all possible rational zeros
    of the function and then find all the actual zeros of: . f(x) \:= \:x^3 + 2x^2 - 21x + 18

    For this problem, the only rational zeros are factors of 18: . \pm1,\:\pm2,\;\pm3,\;\pm6,\:\pm18

    We're in luck . . . The first one works!
    . . f(1) \:=\:1^3 + 2\!\cdot\!1^2 - 21\!\cdot\!1 + 18 \:=\:0
    Hence, (x-1) is a factor of f(x).

    We find that: . f(x)\;=\;(x-1)(x^2+3x-18) \;=\;(x-1)(x-3)(x+6)

    . . Therefore, the zeroes are: . \boxed{x \;=\;1,\,3,\,-6}



    2. Find all the solutions to: . f(x) \:= \:-8x^4 + 128x^2

    Factor: . -8x^2(x^2-16) \:=\:0\quad\Rightarrow\quad-8x^2(x-4)(x+4) \:=\:0

    Therefore, the solutions are: . \boxed{x \;=\;0,\,\pm4}

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  3. #3
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    Thank you for your help! Just one more quick question about the Rational Zero Test...how exactly does it work? Like, if I were to do another similar problem?
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  4. #4
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    if f(x) = a_nx^n+a_{n-1}x^{n-1}+...+a_0
    and all of the coefficients are integers then all rational roots of f can be written as \frac{p}{q} where p|a_0 and q|a_n
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