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June 10th, 2011, 06:33 AM  #1 
Newbie Joined: Jun 2011 Posts: 4 Thanks: 0  Why can't 3/2 be a factor of this polynomial?
Why can't be a factor of ? At first I thought it was due to the combination of 3/2 not matching up with the possibilities given by the rational root test. The possible roots given by the test would be . I don't think that the rational root test gives a complete set of possibilities though, so what other reasons could there be a root can be excluded from being a possible root? 
June 10th, 2011, 06:59 AM  #2 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 141  Re: Why can't 3/2 be a factor of this polynomial?
You've got it. The rational roots theorem says that if p/q is a rational root of some polynomial equation with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient. 3 is not a factor of 5.

June 10th, 2011, 07:12 AM  #3 
Newbie Joined: Jun 2011 Posts: 2 Thanks: 0  Re: Why can't 3/2 be a factor of this polynomial? Rational Root Test Suppose we have a polynomial equation where , the coefficients being integers. If and are nonzero, then each rational solution x, when written as a fraction in lowest terms (i.e., the greatest common divisor of p and q is 1), satisfies p is an integer factor of the constant term , and q is an integer factor of the leading coefficient . The given polynomial may or may not have rational solutions. But cannot be a solution since 3 is not an integer factor of the constant term 5. This is a direct application of rational root test. The proof of the above test can be done using Euclid's Lemma or Gauss's Lemma. 
June 10th, 2011, 07:56 AM  #4 
Newbie Joined: Jun 2011 Posts: 4 Thanks: 0  Re: Why can't 3/2 be a factor of this polynomial?
I understand now, the rational root test gives all rational (hence the name) roots. This just hadn't clicked with me. Thanks for the replies!

June 10th, 2011, 06:04 PM  #5  
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0  Quote:
 
June 12th, 2011, 06:31 AM  #6 
Newbie Joined: Jun 2011 Posts: 4 Thanks: 0  Re: Why can't 3/2 be a factor of this polynomial?
I actually wanted to know why 3/2 couldn't be a root of that polynomial, I just asked the question wrong.


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3 or 2, factor, polynomial 
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