(In reply to Solution (spoiler) - I think
by Kenny M)
Good thinking and a nice approach, Kenny.
But there is a conceptual error. By definition of a+-bi and c+-di, the coefficients a, b, c, d are all real numbers. Therefore, the conclusion that 2 of them must be imaginary is, ipso facto, incorrect.
Here's a slightly different take on the problem:
if f(x) = x^4 - 4x -1, then f'(x) = 4x^3 - 4
So, f'(x) = 0 iff x = +1 => the curve representing the polynomial has only 1 turning point at x = +1.
Now f(-1) = +4 and f(1) = -4 and f(2) = +7
Therefore, the curve is decreasing to the left of x = +1 and increasdng to the right of x = +1.
So, it is concave upwards, and must cross the x-axis at 2 points (i.e.., real numbers) between x = -1 and x = +2.
So, there are precisely 2 real roots.
Edited on August 4, 2011, 8:05 pm