Question

how many solution does x^3-4x+8 have?

Answer 1

Refer to the fundamental theorem of algebra.

Answer 2

The answer here really depends on what is meant by solutions. If you're looking for roots (that is, p, q, and r in (x-p)(x-q)(x-r) = x^3-4x-8), then the answer is always the degree of the polynomial (in this case, 3). If you're looking for distinct roots, you have to account for double roots. For example, x^2-4x+4 has two roots, but they're both 2. So it only has one distinct root. If you're looking for distinct real roots, you have to account for complex roots. For example, x^2+1 has two roots, but no real roots.

In this case, I'm going to guess that you're looking for (possibly distinct) real roots. If you graph the polynomial, you'll see it crosses the x axis once. This means that it has only one distinct real root. I used this method instead of factoring because this polynomial doesn't nicely factor.