Question

What are the possible number of positive real, negative real, and complex zeros of
f(x) = 8x4 + 13x3 – 11x2 + x + 9

Answer 1

There are 2 sign changes so either 2 or 0 positive real roots.

Swapping x for -x there are still 2 sign changes so 2 or 0 negative real roots.

f'(x) = 32x^3 +39x^2 -22x +1
f''(x) = 96x^2 +78x -22 which has positive discriminant so at least 2 real roots

After 0 , function is always increasing so real roots are negative.

Therefore 2 negative real roots and either 2 positive or 2 complex roots.

Answer 2

What are the possible number of positive real, negative real, and complex zeros of
f(x) = 8x4 + 13x3 – 11x2 + x + 9?