Answer
Either 2 or no positive real zeros. No negative zeros.
Work Step by Step
*Refer to page 384 for a complete description of Descarte's Rule of Signs*
Descarte's Rule of Signs applies for two scenarios:
1) Positive real zeros: we identify how many sign changes there are within the polynomial from left to right:
$$f(x) = +3x^{4} -2x^{3} - 8x + 5$$
where we see 2 changes: from positive $3x^{4}$ to negative $2x^{3}$ and from negative $8x$ to positive $5$. This means that, according to the Rule, there are either 2 positive real zeros for this polynomial or $2-2$ = no positive real zeros.
2) Negative real zeros. we substitute $-x$ into the original function:
$$f(-x) = +3(-x)^{4} - 2(-x)^{3} - 8(-x) + 5$$
which simplifies to:
$$f(-x) = +3(1)x^{4} -2(-1)x^{3} - 8(-1)x + 5$$
$$f(-x) = +3x^{4} +2x^{3} + 8x + 5$$
Since there are no sign changes, we can conclude that there are no negative real zeros.
