Difference between revisions of "Rational Root Theorem"

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m (Intermediate)
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===Intermediate===
 
===Intermediate===
  
Find all rational roots of the polynomial <math>x^4-x^3-x^2+x+57</math>
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Find all rational roots of the polynomial <math>x^4-x^3-x^2+x+57</math>.
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Prove that <math>\sqrt{2}</math> is irrational, using the Rational Root Theorem.

Revision as of 02:23, 18 August 2008

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Given a polynomial $P(x) = a_n x^n + a_{n - 1}x^{n - 1} + \ldots + a_1 x + a_0$ with integral coefficients, $a_n \neq 0$. The Rational Root Theorem states that if $P(x)$ has a rational root $r = \pm\frac pq$ with $p, q$ relatively prime positive integers, $p$ is a divisor of $a_0$ and $q$ is a divisor of $a_n$.

As a consequence, every rational root of a monic polynomial with integral coefficients must be integral.

This gives us a relatively quick process to find all "nice" roots of a given polynomial, since given the coefficients we have only a finite number of rational numbers to check.

Problems

Intermediate

Find all rational roots of the polynomial $x^4-x^3-x^2+x+57$.

Prove that $\sqrt{2}$ is irrational, using the Rational Root Theorem.

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