Answer
The maximum number of real zeros of $4$.
The number of positive real zeros is $1$
The number of negative real zeros is $1$.
Work Step by Step
We should remember that the maximum number of zeros of a polynomial function $f(x)$ can not be greater than its degree.
We will consider Descartes' Rule of Signs for explaining this solution:
(1) The number of negative real zeros of a polynomial function $f(x)$ is either equal to the number of variations in the sign of the non-zero coefficients of $f(−x)$ or that number minus an even integer.
(2) The number of positive real zeros of a polynomial function $f(x)$ is either equal to the number of variations in the sign of the non-zero coefficients of $f(x)$ or that number minus an even integer.
We can notice from the given polynomial function that the highest degree is $4$, so the maximum number of real zeros is $4$.
$f(x)=5x^4+2x^2-6x-5$ has 1 variation in the sign. So, the number of positive real zeros is $1$.
and $f(−x)=5x^4+2x^2+6x-5$ has 1 variation in the sign. So, the number of negative real zeros is $1$.
