For the equation, 2x^4 - 5x^3+10=0 find the number of complex roots and the possible number of real roots.

A. 3 complex roots; 0, 2 or 4 real roots

B. 3 complex roots; 1 or 3 real roots

C. 4 complex roots; 0, 2 or 4 real roots

D. 4 complex roots; 1 or 3 real roots

The given equation is

2x⁴ - 5x³ + 10 = 0

Because this is a 4th degree polynomial, there are 4 possible real and complex roots.

Let f (x) = 2x⁴ - 5x³ + 10

Apply Descartes' Rule of signs.

There are 2 sign changes, so there are 2 possible positive real roots.

f (-x) = 2x⁴ + 5x³ + 10

There are no sign changes, so there are no negative real roots.

This means that

(a) There are possibly 2 real positive roots, and a conjugate pair of 2 complex roots;

(b) 0 real roots, and 2 pairs of 4 complex roots.

Of the given answers, only C. can be correct.

Note that complex roots always occur as conjugate pairs, so A and B are incorrect.

Answer: C

4 complex roots; 0, 2, or 4 real roots