Theory of EquationsmediumFree

Theory of Equations: Let Roots Real Non Real Value Equal

JEE Maths question with a full step-by-step solution.

Question

Let

\alpha, \beta, \gamma, \delta

be the roots (real or non-real) of

x^4 - 3x + 1 = 0

. The value of

\alpha^3 + \beta^3 + \gamma^3 + \delta^3

is equal to

6

9

correct

12

15

Solution

Step 1: From

x^4 = 3x - 1

(rearranging), divide by

x

x^3 = 3 - \dfrac{1}{x}

for each root. Step 2:

\sum \alpha^3 = 4 \cdot 3 - \sum \frac{1}{\alpha} = 12 - \frac{\sum \alpha\beta\gamma}{\alpha\beta\gamma\delta}

Step 3: By Vieta's for

x^4 + 0x^3 + 0x^2 - 3x + 1 = 0

\sum\alpha\beta\gamma = 3

and

\alpha\beta\gamma\delta = 1

\sum \alpha^3 = 12 - \frac{3}{1} = 9

Correct answer: (2)

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