Question

How do you find all possible rational zeros of `f(x) = 2x^3 - 5x^2 + 3x - 1`?

Answer 1

Use the rational root theorem to help find that it has no rational zeros.

Explanation:

`f(x) = 2x^3-5x^2+3x-1`

By the rational root theorem, any rational zeros of `f(x)` are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `-1` and `q` a divisor of the coefficient `2` of the leading term.

That means that the only possible rational zeros are:

`+-1/2`, `+-1`

In addition, note that there are no changes of signs of coefficients in `f(-x) = -2x^3-5x^2-3x-1`, so `f(x)` has no negative zeros.

That leaves possible rational zeros:

`1/2`, `1`

Then we find:

`f(1/2) = 1/4-5/4+3/2-1 = -1/2`

`f(1) = 2-5+3-1 = -1`

So this cubic has no rational zeros.