this is not a homework question, but rather I feel like there is a contradiction in the theorem and just want clarity. I know the theorem is correct so Im looking for help in where the mistake is in my logic. 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!