Another Princeton Review error

Forum for the GRE subject test in mathematics.
joey
Posts: 32
Joined: Fri Oct 16, 2009 3:53 pm

Another Princeton Review error

Postby joey » Fri Oct 16, 2009 4:13 pm

At the top of p.15 in the 3rd edition, it states: "[I]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?

User avatar
lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am

Re: Another Princeton Review error

Postby lime » Sat Oct 17, 2009 2:26 am

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.

joey
Posts: 32
Joined: Fri Oct 16, 2009 3:53 pm

Re: Another Princeton Review error

Postby joey » Sat Oct 17, 2009 8:35 am

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.

User avatar
lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am

Re: Another Princeton Review error

Postby lime » Sat Oct 17, 2009 9:10 am

Agree. The name "irrational roots theorem" sounds a bit misleading tending to make generalization.




Return to “Mathematics GRE Forum: The GRE Subject Test in Mathematics”



Who is online

Users browsing this forum: Bing [Bot], marmle, omhk and 3 guests