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Math Help - Third degree ploynomial question : 2

  1. #1
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    Third degree ploynomial question : 2

    if a,b,c are the roots of the equation: x^3 + 2x^2 + 3x + 3 = 0 then the value of

    [a/(a+1)]^3 + [b/(b+1)]^3 + [c/(c+1)]^3 is?
    ----------

    I know that this equation has one real root and two imaginary roots. the real root lies between -2 and 0. But unable to proceed further.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let y_1=\frac{a}{a+1}, \ y_2=\frac{b}{b+1}, \ y_3=\frac{c}{c+1}

    We form the equation with the roots y_1, \ y_2, \ y_3

    y=\frac{x}{x+1}\Rightarrow x=\frac{y}{1-y}

    Replace x in the x-equation an we get the equation in y: y^3-5y^2+6y-3=0

    S_1=y_1+y_2+y_3=5

    S_2=y_1^2+y_2^2+y_3^2=(y_1+y_2+y_3)^2-2(y_1y_2+y_1y_3+y_2y_3)=25-12=13

    Let S_3=y_1^3+y_2^3+y_3^3

    We have

    y_1^3-5y_1^2+6y_1-3=0
    y_2^3-5y_2^2+6y_2-3=0
    y_3^3-5y_3^2+6y_3-3=0

    Adding the three relations we have

    S_3-5S_2+6S_1-9=0\Rightarrow S_3=5S_2-6S_1+9=65-30+9=44
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  3. #3
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    thanks for that solution.

    can u please tell me some good resource on the net to practice solved question on third degree polynomials? I cant find good resources..
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