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Math Help - Zeros of a cubic function

  1. #1
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    Zeros of a cubic function

    Hello,

    I'm trying to help my brother with a math problem, but I just haven't done this stuff in quite a while so I don't even know where to start. I'd appreciate any assistance!

    Cheers.

    Find the zeros for the function f(x) = ax^3 + bx^2 -106x + d, given that f(-1) = 343, f(2) = 40, and f(3) = -33.
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  2. #2
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    So I started an attempt at this by solving for the given values, then reducing two pairs of equations down to two equations with two unknowns (I eliminated d). Then I substituted b as a function of a from one equation into the other, but I got a = -(8/14), which leads me to believe I made a mistake (this is a high school question, so I expect numbers to work out nicely).
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  3. #3
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    Quote Originally Posted by RandomGuy8181 View Post
    Hello,

    I'm trying to help my brother with a math problem, but I just haven't done this stuff in quite a while so I don't even know where to start. I'd appreciate any assistance!

    Cheers.

    Find the zeros for the function f(x) = ax^3 + bx^2 -106x + d, given that f(-1) = 343, f(2) = 40, and f(3) = -33.
     <br />
f(x) = ax^3+bx^2-106x+d<br />

    Use, f(-1) = 343, we got

    343  = a(-1)^3+b(-1)^2-106(-1) +d

     \Rightarrow 343= -a+b+106+d

     \Rightarrow  -a+b+d=237 ..................(1)

    Now, Use, f(2) = 40, we got

    40  = a(2)^3+b(2)^2-106(2) +d

     \Rightarrow 343= 8a+4b-212+d

     \Rightarrow  8a+4b+d=555 ..................(2)

    By Using, f(3) = -33, we got

    33  = a(3)^3+b(3)^2-106(3) +d

     \Rightarrow  27a+9b+d=351 ..................(3)

    Now, subtract eqn (1) from eqn(2) and eqn(3),
    that means, (2) - (1) and (3) - (1), we got

    9a + 3b = 318 .........................(4)

    28a + 8b = 114 .......................(5)

    divide eqn(4) by 3 and eqn (5) by 2

    3a + b = 106

    14a + 4b = 57

    Now solve these two eqns and find a and b. then put those values of a and b in eqn (1) to find d. Then you will have a cubic function.
    Now, after that use factor theorem to find one factor. Then divide the cubic function with that factor to find a quadratic factor. Then factor that quadratic function. Finish it.
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