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Math Help - advanced functions

  1. #1
    Senior Member polymerase's Avatar
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    Exclamation advanced functions

    A cubic polynomial has x=-3 as one of its roots and y-intercept 9. If the graph passes through (-2,3) and (1,16), find the equation of the polynomial.
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  2. #2
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    You can use the given info to build a system and then solve for the coefficients.

    The y-intercept is 9, therefore, x=0:

    a(0)^{3}+b(0)^{2}+c(0)+d=9

    A root is -3:

    a(-3)^{3}+b(-3)^{2}+c(-3)+d=0

    Passes through (-2,3):

    a(-2)^{3}+b(-2)^{2}+c(-2)+d=3

    Passes through (1,16):

    a(1)^{3}+b(1)^{2}+c(1)+d=16

    Now, you have 4 equations with 4 unknowns:

    \left[\begin{array}{cccc|c}0&0&0&1&9\\-27&9&-3&1&0\\-8&4&-2&1&3\\1&1&1&1&16\end{array}\right]

    Solve the system and you get:

    a=\frac{1}{3}, \;\ b=\frac{5}{3}, \;\ c=5, \;\ d=9

    Therefore, the polynomial is:

    \boxed{\frac{1}{3}x^{3}+\frac{5}{3}x^{2}+5x+9}

    This is one way to approach these. Perhaps try this on your other post regarding the quartic.
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