Question

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

Answer 1

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