*rrational roots of rational-coefficient polynomial equations must occur in conjugate radical pairs." However, this fails to account for irrational roots not of the form $$s+t\sqrt{u}$$.*

Following their line of reasoning may prove disastrous. For instance, question #17 of GR0568 asks to find the number of real roots of $$p(x)=2x^5+8x-7$$. Since $$p(0)<0$$ and $$p(1)>0$$, there is at least one root. Moreover, that root must be irrational, as a quick application of the rational roots theorem shows. If we take PR at their word, we're forced to conclude that there are at least two zeroes. However, $$p'(x)>0$$, so there can be at most one zero. Indeed, the answer key confirms there is exactly one.

Studying with this guide has been nothing but a crap shoot. It seems I'm only slightly more likely to learn useful information than I am to fall prey to subtle yet fatal errors. After two revisions, obvious pitfalls remain. What is going on here?

Following their line of reasoning may prove disastrous. For instance, question #17 of GR0568 asks to find the number of real roots of $$p(x)=2x^5+8x-7$$. Since $$p(0)<0$$ and $$p(1)>0$$, there is at least one root. Moreover, that root must be irrational, as a quick application of the rational roots theorem shows. If we take PR at their word, we're forced to conclude that there are at least two zeroes. However, $$p'(x)>0$$, so there can be at most one zero. Indeed, the answer key confirms there is exactly one.

Studying with this guide has been nothing but a crap shoot. It seems I'm only slightly more likely to learn useful information than I am to fall prey to subtle yet fatal errors. After two revisions, obvious pitfalls remain. What is going on here?