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Math Help - Determine an equation....

  1. #1
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    Determine an equation....

    ... of a polynomial function with zeros at 2,-2, 1 and a y-intercept of 24

    I put (x-2)(x-1)(x+2) +24 and factored out but it didn't work correctly

    I got x3-1x2-4x+24

    What am I doing wrong here?
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  2. #2
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    Re: Determine an equation....

    Quote Originally Posted by rhickman View Post
    ... of a polynomial function with zeros at 2,-2, 1 and a y-intercept of 24

    I put (x-2)(x-1)(x+2) +24 and factored out but it didn't work correctly

    I got x3-1x2-4x+24

    What am I doing wrong here?
    you can't just add 24 and expect the zeros to remain the same.

    What you can do is scale the entire thing

    $A(x-2)(x-1)(x+2)$

    Now find A so that the above expression is 24 when x=0.

    $A(x-2)(x-1)(x+2)|_{x=0}=24$

    $A(-2)(-1)(2)=24$

    $4A=24$

    $A=6$

    so

    $6(x-2)(x-1)(x+2)$ has the properties you want.
    Thanks from prasum
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  3. #3
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    Re: Determine an equation....

    If the polynomial has zeroes at 2, - 2, and 1, then it is of the form

    $f(x) = a(x - 2)^b(x + 2)^c(x - 1)^d = 0,\ where\ b,\ c,\ d \in \mathbb Z^+\ and\ a \ne 0.$

    An infinite number of polynomials in that form have a y-intercept of 24.

    But there is only one cubic of that form with y-intercept of 24.

    $f(x) = a(x - 2)(x + 2)(x - 1) = 0\ and\ f(0) = 24 \implies a(x^2 - 4)(x - 1) = 0 \implies a(x^3 - x^2 - 4x + 4) = 0 \implies$

    $4a = 24 \implies a = 6 \implies f(x) = 6x^3 - 6x^2 - 24x + 24.$

    $f(-2) = 6(-8) - 6(4) - 24(-2) + 24 = -48 - 24 + 48 + 24 = 0.$ Check

    $f(2) = 6(8) - 6(4) - 24(2) + 24 = 48 - 24 - 48 + 24 = 0.$ Check

    $f(1) = 6(1) - 6(1) - 24(1) + 24 = 6 - 6 - 24 + 24 = 0.$ Check

    $f(0) = 6(0) - 6(0) - 24(0) + 24 = 0 + 0 + 0 + 24 = 24.$ Check
    Last edited by JeffM; May 19th 2014 at 09:22 AM. Reason: typos (sigh)
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