# Prove there are no rational solutions

• Apr 18th 2010, 02:49 PM
treetheta
Prove there are no rational solutions
Prove that there are no rational solutions to the equation x^3 + x^2 = 1
• Apr 18th 2010, 04:19 PM
Laurent
Quote:

Originally Posted by treetheta
Prove that there are no rational solutions to the equation x^3 + x^2 = 1

Let $x=\frac{p}{q}$ with $p\wedge q=1$; you get $p^3+qp^2=q^3$ hence (...) $q|p$ and thus $q=1$, which mean $x$ is an integer. I let you fill in the (quick) dots and handle it from there (beware, $x^3$ may be negative).
• Apr 18th 2010, 07:17 PM
Drexel28
Quote:

Originally Posted by treetheta
Prove that there are no rational solutions to the equation x^3 + x^2 = 1

Or, maybe the rational root theorem...