Question

Solve the given polynomial equation. Use the Rational Zero Theorem and​ Descartes's Rule of Signs as an aid in obtaining the first root.
2 x cubed minus 5 x squared minus 5 x minus 1 equals 02x3−5x2−5x−1=0

Answer 1

2x^3 - 5x^2 - 5x - 1 = 0

let f(x) = 2x^3 - 5x^2 - 5x - 1
try f(1) = 2 - 5 - 5 - 1 ≠ 0
try f(-1) = -2 - 5 + 5 - 1 ≠ 0
try f(1/2) = 1/4 - 5/4 - 5/2 - 1≠ 0
try f(-1/2) = -1/4 - 5/4 + 5/2 - 1 = 0 , yeahh!
so (2x + 1) is a factor
and x = -1/2

by long algebraic division, which I assume you know how to do
2x^3 - 5x^2 - 5x - 1 = (2x+1)(x^2 - 3x - 1)

so for the other two roots:
x^2 - 3x - 1 = 0
x = (3 ± √13)/2

Answer 2

the rule of signs says that there is at most one positive root, and 3 negative roots.

rational zeros states that any rational root will be ±1 or ±1/2