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June 10th, 2011, 06:33 AM   #1
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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?
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June 10th, 2011, 06:59 AM   #2
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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.
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June 10th, 2011, 07:12 AM   #3
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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.
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June 10th, 2011, 07:56 AM   #4
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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!
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June 10th, 2011, 06:04 PM   #5
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Quote:
Originally Posted by figgy
Why can't be a factor of ?
I assume you're asking why can't x - 3/2 be a factor of the polynomial? Because 3/2 is clearly can be a factor of the polynomial.
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June 12th, 2011, 06:31 AM   #6
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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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