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Another Princeton Review error

Posted: Fri Oct 16, 2009 4:13 pm
by joey
At the top of p.15 in the 3rd edition, it states: "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?

Re: Another Princeton Review error

Posted: Sat Oct 17, 2009 2:26 am
by lime
There is no contradiction here. The theorem concerns only roots in form
$$a+\sqrt{b}$$
while given equation has roots in form
$$a + \sqrt{b} + \sqrt[5]{c}$$
for which theorem does not apply.

Re: Another Princeton Review error

Posted: Sat Oct 17, 2009 8:35 am
by joey
Agreed. The problem is not with the theorem, but with Princeton Review's summary of it. We're led to believe that all irrational roots of polynomials take the form
$$s+t\sqrt{u}$$
and appear in conjugate pairs. Obviously, this is false, as demonstrated here.

Re: Another Princeton Review error

Posted: Sat Oct 17, 2009 9:10 am
by lime
Agree. The name "irrational roots theorem" sounds a bit misleading tending to make generalization.