Question

Is there a number that is exactly 4 more than its cube?

Answer 1

x - x^3 = 4

Answer 2

Well, lets call such a number k.

That would imply that \[k^3 + 4 = k\]\[\implies k^3 - k + 4 = 0\]
So now we just need to find the roots of this polynomial.

Answer 3

It doesn't look like it'll be a terribly pretty number, but since it's an odd degree polynomial it must have at least one root, so the number does exist.

Answer 4

It won't. All three roots are irrational.

Answer 5

We can find the exact solutions for this equations. They involve cube roots of quadratic surds.