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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 $3/2$ be a factor of $4x^5+cx^3-dx+5=0$? 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 $\pm 1/4, 1/2, 1, 5/4, 5$. 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: 140 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 $a_nx^n+a_{n-1}x^{n-1}+...+a_0$ where $a_n\neq 0$, the coefficients being integers. If $a_0$ and $a_n$ are nonzero, then each rational solution x, when written as a fraction $x= p/q(q\neq0)$ 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 $a_0$, and q is an integer factor of the leading coefficient $a_n$. The given polynomial $4x^5+cx^3-dx+5=0$ may or may not have rational solutions. But $3/2$ 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
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Quote:
 Originally Posted by figgy Why can't $3/2$ be a factor of $4x^5+cx^3-dx+5=0$?
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.

 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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