Question

If α, β, γ and δ are the roots of $x^4$ + 2 $x^3$ + 3 $x^2$ + 4x + 5 = 0. Find the equation whose roots are $\frac{1}{\alpha }$, $\frac{1}{\beta }$, $\frac{1}{\gamma }$, $\frac{1}{\delta }$

• A. x⁴ - 2x³ + 3x² - 4x + 5 = 0
• B. x⁴ + 3x³ + 2x² + 4x + 5 = 0
• C. 5x⁴ + 4x³ + 3x² + 2x + 1 = 0
• D. 5x⁴ - 4x³ + 3x² - 2x + 1 = 0

Answer 1

The correct option is C

5x⁴ + 4x³ + 3x² + 2x + 1 = 0

Roots of $x^4$ + 2 $x^3$ + 2 $x^2$ + 4x + 5 = 0 ---------------------(1) are α, β, γ, δ

Changing x to $\frac{1}{x}$

$\frac{1}{x^4}$ + $\frac{2}{x^3}$ + $\frac{3}{x^2}$ + $\frac{4}{x}$ + 5 = 0

5 $x^4$ + 4 $x^3$ + 3 $x^2$ + 2x + 1 = 0 _________(2)

Roots of the equation 2 being the reciprocal of equation 1.

$\frac{1}{\alpha }$, $\frac{1}{\beta }$, $\frac{1}{\gamma }$, $\frac{1}{\delta }$