The sum of the fourth powers of the roots of | vedclass

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(D) Let the roots of the equation $x^3+x+1=0$ be $\alpha, \beta, \gamma$. Let $S_n = \alpha^n + \beta^n + \gamma^n$.By Newton's Sums for the equation

Vedclass · Accepted Answer

$2$

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(D) Let the roots of the equation $x^3+x+1=0$ be $\alpha, \beta, \gamma$. Let $S_n = \alpha^n + \beta^n + \gamma^n$.By Newton's Sums for the equation $x^3+p_1x^2+p_2x+p_3=0$,where $p_1=0, p_2=1, p_3=1$:$S_1 + p_1 = 0$ $\Rightarrow S_1 + 0 = 0$ $\Rightarrow S_1 = 0$.$S_2 + p_1S_1 + 2p_2 = 0$ $\Rightarrow S_2 + 0(0) + 2(1) = 0$ $\Rightarrow S_2 = -2$.$S_3 + p_1S_2 + p_2S_1 + 3p_3 = 0$ $\Rightarrow S_3 + 0(-2) + 1(0) + 3(1) = 0$ $\Rightarrow S_3 = -3$.$S_4 + p_1S_3 + p_2S_2 + p_3S_1 = 0 \Rightarrow S_4 + 0(-3) + 1(-2) + 1(0) = 0$.$S_4 - 2 = 0 \Rightarrow S_4 = 2$.

The sum of the fourth powers of the roots of the equation $x^3+x+1=0$ is

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