Question

Find the equation whose roots are equal in magnitude but opposite in sign to the roots of the equation $x^5$- 3 $x^3$ + 2 $x^2$ + 4x + 1 = 0.

• A. x⁵ - 3x³ + 2x² + 4x + 1 = 0
• B. x⁵ + x³ + x² + x + 1 = 0
• C. x⁵ - 3x³ - 2x² + 4x - 1 = 0
• D. x⁵ + 2x³ + 3x² + 5x - 1 = 0

Answer 1

The correct option is C

x⁵ - 3x³ - 2x² + 4x - 1 = 0

Solution: $x^5$- 3 $x^3$ + 2 $x^2$ + 4x + 1 = 0 ____ (1)

Let x be a root of the given equation and y a root of the required equation.

Then y = - x or x = - y

Put x = - y in the equation 1, we have

- $y^5$ + 3 $y^3$ + 2 $y^2$ - 4y + 1 = 0

Express it in terms of x

- $x^5$ + 3 $x^3$ + 2 $x^2$ - 4x + 1 = 0

$x^5$ - 3 $x^3$ - 2 $x^2$ + 4x - 1 = 0