Answer
$1$ positive root, $1$ negative root
$2$ real roots
Work Step by Step
Descartes' Rule of Signs states that the possible number of the positive roots of a polynomial is equal to the number of sign changes in the coefficients or less than the number of sign changes by a multiple of $2$ AND the possible number of negative roots of a polynomial is equal to the number of sign changes or less than the total number of sign changes by a multiple of $2$ after substituting $−x$ for $x$. The substitution has the effect of negating all of the odd-power terms in the polynomial.
Therefore,
$P(x)=2x^6+5x^4-x^3-5x-1$, has $1$ sign change, between $5x^4$ and $-x^3$. Therefore, it has $1$ positive root.
$P(-x)=2x^6+5x^4+x^3+5x-1$, has $1$ sign change, between $5x$ and $-1$. Therefore, it has $1$ negative root.
Therefore, It has a total of $2$ real roots.
