The sum of the cubes of all the roots of the equation x? ? 3x³ ? 2x² + 3x + 1 = 0 is

Question

The sum of the cubes of all the roots of the equation $x^{4} - 3 x^{3} - 2 x^{2} + 3 x + 1 = 0$ is _____

Smart Achievers · Accepted Answer

(36)

$x^{4} - 3 x^{3} - x^{2} - x^{2} + 3 x + 1 = 0$

$(x^{2} - 1) (x^{2} - 3 x - 1) = 0$

$Let the roots of x^{2} - 3 x - 1 = 0 be α and β . Other two roots of the given equation are 1 and - 1 .$

$So, sum of cubes of roots = 1^{3} + {(- 1)}^{3} + α^{3} + β^{3}$

$= {(α + β)}^{3} - 3 α β (α + β) = {(3)}^{3} - 3 (- 1) (3) = 36$

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