JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Quadratic Equations and Inequalities

The sum of the cubes of all the roots of the equation

x^{4} - 3 x^{3} - 2 x^{2} + 3 x + 1 = 10

is ______.

Answer: 36

Solution

Bringing everything to one side,

x^{4} - 3 x^{3} - 2 x^{2} + 3 x - 9 = 0

. Rather than solving for the roots individually, read the elementary symmetric functions straight off the coefficients:

e_{1} = 3, e_{2} = - 2, e_{3} = - 3, e_{4} = - 9

. Newton's identities then give the power sums in turn:

p_{1} = e_{1} = 3

;

p_{2} = e_{1} p_{1} - 2 e_{2} = 3 \cdot 3 - 2 (- 2) = 13

; and

p_{3} = e_{1} p_{2} - e_{2} p_{1} + 3 e_{3} = 3 \cdot 13 - (- 2) \cdot 3 + 3 (- 3) = 39 + 6 - 9 = 36

. The sum of the cubes of all the roots is

36

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Integer answer