Question

list all of the possible rational zeros of f(x) = 3x^5 - 7x^3 + 2x - 15.

Answer 1

Let the rational roots (if any) be represented by p/q, where p, q ∈ℤ, p≠0, q≠0, and where p|-15 (constant term) and q|3 (leading coefficient).

Note: a|b means a divides b.

The possible candidates for rational roots are therefore:

±{1,3,5}/±{1,3}
or
±1, ±3, ±5, ±1/3, and ±5/3.

Evaluating f(x) for each of the candidates confirm that there are no rational roots.

---- extra ----

To find the real zeroes, we need to find the critical points (where f'(x) = 0) and evaluate f(x) at these points.

At all four critical points f'(x)<0, therefore there is only one real zero between 1.14 and infinity.
The single real (irrational) zero is at x=1.747 approx.