Question

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

Answer 1

All of the roots are rational: `-1/2`, `-1`, and `1`.

Explanation:

Usually, it is hard to find the zeros of third-degree polynomials. However, for this one, we can do so by a special trick.

We can rewrite this polynomial as `2x^3+x^2-2x-1=x^2(2x+1)-(2x+1)`. We can then factor `2x+1` from this, getting `(2x+1)(x^2-1)`. We recall that `a^2-b^2=(a+b)(a-b)` to further factor `(x^2-1)` into `(x+1)(x-1)`. This polynomial then becomes `(2x+1)(x+1)(x-1)`. The roots are therefore `-1/2`, `-1`, and `1`. All of these are rational.