Find the equation whose roots are equal in magnitude but opposite in sign to the roots of the equation x5- 3 x3 + 2 x2 + 4x + 1 = 0.
x5 - 3x3 - 2x2 + 4x - 1 = 0
Solution: x5- 3 x3 + 2 x2 + 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
- y5 + 3 y3 + 2 y2 - 4y + 1 = 0
Express it in terms of x
- x5 + 3 x3 + 2 x2 - 4x + 1 = 0
x5 - 3 x3 - 2 x2 + 4x - 1 = 0