Answer
$3$ or $1$ positive roots
$3$ or $1$ 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,
$P(x)=2x^3-x^2+4x-7$, has $3$ sign changes, between $2x^3$ and $-x^2$, between $-x^2$ and $4x$, and between $4x$ and $-7$. Therefore, has $3$ or $1$ positive roots.
$P(-x)=-2x^3-x^2-4x-7$, has no sign change. Therefore, it has no negative roots.
Therefore, It has a total of $3$ or $1$ real roots.
