Question

use descartes rule of signs to find how many possible positive real roots this polynomial has:
$p(x) = -2x^5 + 3x^4 - 9x^3 + x^2 - 7x$
0
4, 2 or 0
1

Explanation:

Step1: List polynomial coefficients

$-2, +3, -9, +1, -7$

Step2: Count sign changes

1. $-2$ to $+3$: sign change
2. $+3$ to $-9$: sign change
3. $-9$ to $+1$: sign change
4. $+1$ to $-7$: sign change

Total sign changes: 4

Step3: Apply Descartes Rule

Subtract even integers (2, 4, ...) from the total sign changes to get all possible counts of positive real roots: $4$, $4-2=2$, $2-2=0$.

Answer:

4, 2 or 0