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Math Help - Cubic function with integral coefficients.

  1. #1
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    Cubic function with integral coefficients.

    "A cubic function f(x) with integral coefficients has the following properties: f(3/2) = 0, (x-2) is a factor of f(x), and f(4) = 50. Determine f(x)."

    How do I solve this?

    The two factors are (x - 3/2) and (x - 2), right?

    I don't know what to do from there.
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  2. #2
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    Quote Originally Posted by Jeavus View Post
    "A cubic function f(x) with integral coefficients has the following properties: f(3/2) = 0, (x-2) is a factor of f(x), and f(4) = 50. Determine f(x)."

    How do I solve this?

    The two factors are (x - 3/2) and (x - 2), right?

    I don't know what to do from there.
    So,
    f(x) = A(x-3/2)(x-2)(x-c)
    And f(4)=50 so,
    50 = A(4-3/2)(4-2)(4-c)
    It seems you need another condition.
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  3. #3
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    Hello, Jeavus!

    A cubic function f(x) with integral coefficients has the following properties:
    . . f\left(\frac{3}{2}\right) \:=\: 0,\;\;(x-2) is a factor of f(x), and f(4) \:= \:50
    Determine f(x).

    Two of the factors are: . (x - 2) and (2x-3)

    Since the function is a cubic, there is a third factor: . (x - a)

    Hence: . f(x) \;\;=\;\;(x-2)(2x-3)(x-a) \;\;=\;\;2x^3 - (2a+7)x^2 + (7a+6)x - 6a


    Since f(4) = 50, we have: . 2\!\cdot\!4^3 - (2a+7)\!\cdot\!4^2 + (7a+6)\!\cdot\!4 - 6a\;=\;50

    . . which simplifies to: . a \:=\:-1


    Therefore, the cubic is: . {\color{blue}f(x)\;=\;2x^3 - 5x^2 - x + 6}

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