First slide

Theory of expressions

Question

The maximum possible number of real roots of equation $x^{5} - 6 x^{2} - 4 x + 5 = 0$ is

Moderate

A

0
B

3
C

4
D

5

Solution

Let $f (x) = x^{5} - 6 x^{2} - 4 x + 5 = 0$
Then the number of change of sign in f(x) is 2 therefore f(x) can have at most two positive real roots.
now, $f (- x) = - x^{5} - 6 x^{4} + 4 x + 5 = 0$
Then the number of change of sign is 1.
Hence f(x) can have at most one negative real root.
So that total possible number of real roots is 3.

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