# What are all the possible rational zeros for f(x)=2x^3+x^2-2x-1 and how do you find all zeros?

All of the roots are rational: $- \frac{1}{2}$, $- 1$, and $1$.
We can rewrite this polynomial as $2 {x}^{3} + {x}^{2} - 2 x - 1 = {x}^{2} \left(2 x + 1\right) - \left(2 x + 1\right)$. We can then factor $2 x + 1$ from this, getting $\left(2 x + 1\right) \left({x}^{2} - 1\right)$. We recall that ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$ to further factor $\left({x}^{2} - 1\right)$ into $\left(x + 1\right) \left(x - 1\right)$. This polynomial then becomes $\left(2 x + 1\right) \left(x + 1\right) \left(x - 1\right)$. The roots are therefore $- \frac{1}{2}$, $- 1$, and $1$. All of these are rational.