Question

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5x^3 - 4x^2 + 3x - 2 = 0$, then $\alpha^3 + \beta^3 + \gamma^3$ equals

• A. $\dfrac{17}{25}$
• B. $\dfrac{394}{125}$
• C. $\dfrac{34}{125}$
• D. $\dfrac{34}{25}$

Hint:

Use identities for cubes of roots and express in terms of elementary symmetric functions for polynomial root problems.

Answer 1

The Correct Option is C

Use identity: $\alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)^3 - 3(\alpha + \beta + \gamma)(\alpha \beta + \beta \gamma + \gamma \alpha) + 3\alpha \beta \gamma$
From the equation: $a = 5, b = -4, c = 3, d = -2$
Sum of roots: $\dfrac{4}{5}$, sum of product of roots two at a time: $\dfrac{3}{5}$, product: $\dfrac{2}{5}$
Now, \[ \alpha^3 + \beta^3 + \gamma^3 = \left(\frac{4}{5}\right)^3 - 3. \frac{4}{5} . \frac{3}{5} + 3 . \frac{2}{5} = \frac{64}{125} - \frac{36}{25} + \frac{6}{5} \] Simplify: \[ \frac{64}{125} - \frac{180}{125} + \frac{150}{125} = \frac{34}{125} \]