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Math Help - Polynomial Funtion HELP PLEASE!!!

  1. #1
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    Polynomial Funtion HELP PLEASE!!!



    Let f(x) be a polynomial function such that, for all real x.

    f(x^2 +2) = x^4 + 10x^2 + 4.

    Evaluate f(x^2 - 2).


    THANKS FOR YOUR HELP!!!!!!
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  2. #2
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    Quote Originally Posted by rlarach View Post


    Let f(x) be a polynomial function such that, for all real x.

    f(x^2 +2) = x^4 + 10x^2 + 4.

    Evaluate f(x^2 - 2).


    THANKS FOR YOUR HELP!!!!!!
    Well, the "brute force" method would be:
    f(x^2 + 2) = x^4 + 10x^2 + 4

    Let y = x^2 + 2.

    Solve for x:
    x^2 = y - 2

    x = \pm \sqrt{y - 2}

    Ignoring the difficulty of the \pm let's blithely continue:
    f(y) = ( \pm \sqrt{y - 2} )^4 + 10 ( \pm \sqrt{y - 2} ) + 4

    f(y) = (y - 2)^2 + 10(y - 2) + 4

    Now let y = x^2 - 2.

    f(x^2 - 2) = (x^2 - 2 - 2)^2 + 10(x^2 - 2 - 2) + 4

    f(x^2 - 2) = (x^2 - 4)^2 + 10(x^2 - 4) + 4

    which you can expand, or not, as you choose.

    -Dan
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  3. #3
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    Quote Originally Posted by rlarach View Post


    Let f(x) be a polynomial function such that, for all real x.

    f(x^2 +2) = x^4 + 10x^2 + 4.

    Evaluate f(x^2 - 2).


    THANKS FOR YOUR HELP!!!!!!
    <br />
f(x)=x^2+6x-12<br />

    RonL
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  4. #4
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    How Do You Get X^2 + 6x - 12
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  5. #5
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    Quote Originally Posted by rlarach View Post
    How Do You Get X^2 + 6x - 12
    By inspection, the required function is obviously a quadraic with leading
    coefficient 1, so is of the form:

    f(x)=x^2+bx+c.

    Then:

    f(x^2+2)=(x^2+2)^2+b(x^2+2)+c = x^4+10x^2+4

    Now expend the middle term and equate coefficients.

    RonL
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