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December 15th, 2016, 12:24 PM   #1
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alpha and beta roots!

α (alpha) and β (beta) are two of the roots of x^3 + ax^2 + bx + c = 0.
Prove that αβ is a root of x^3 - bx^2 + acx - c^2 = 0.

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December 15th, 2016, 05:05 PM   #2
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I suspect that the Fundamental Theorem of Algebra will help you out.

$(x - \alpha )( x - \beta ) ( x - \gamma ) = x^3 - ( \alpha + \beta + \gamma ) x^2 + ( \alpha \beta + \alpha \gamma + \beta \gamma)x - \alpha \beta \gamma = x^3 + ax^2 + bx + c \implies$

$\alpha \beta \gamma =-\ c\ and\ \alpha \beta + \alpha \gamma + \beta \gamma = b\ and\ \alpha + \beta + \gamma = -\ a.$
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