Question

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

Answer 1

Answer :

Let f(x) = 2x⁴ - 5x³ + 10

f(x) has 2 sign changes. So, by Descartes Rule of Signs, f(x) has either 2 positive roots or none.

f(-x) = 2(-x)⁴ - 5(-x)³ + 10 = 2x⁴ + 5x³ + 10

f(-x) has no sign changes. So, by Descartes Rule of Signs, f(x) has no negative roots.

f(x) has degree 4. So, counting multiplicities, there are a total of 4 roots.

First possibility: The equation may have 2 positive roots , 0 negative roots and 2 imaginary roots

Second possibility: The equation may have 0 positive ,0 negative and 4 imaginary roots .