take f(x) = x^3 + x^2 - 4x- 7

the rational roots theorem says if there are any rational roots they must be in the set: plus or minus 7 and 1. none of which work using synthetic division.

**so the logic would dictate there should be no rational roots.**

however, when you graph it you can see there is one root. but if this root were irrational, I believe the conjugate is always a root - hence there should be two. so therefore it cannot be an irrational root?

**therefore it must be rational?**

which contradicts my previous statement from the rational roots theorem stating there were no rational zeros in this function.

hence my confusion...can someone clarify where I made my mistake in my logic or which assumption I made that was wrong? thanks!