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| − | {{stub}}
| + | #REDIRECT[[Rational root theorem]] |
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| − | Given a [[polynomial]] <math>P(x) = a_n x^n + a_{n - 1}x^{n - 1} + \ldots + a_1 x + a_0</math> with [[integer | integral]] [[coefficient]]s, <math>a_n \neq 0</math>. The '''Rational Root Theorem''' states that if <math>P(x)</math> has a [[rational number| rational]] [[root]] <math>r = \pm\frac pq</math> with <math>p, q</math> [[relatively prime]] [[positive integer]]s, <math>p</math> is a [[divisor]] of <math>a_0</math> and <math>q</math> is a divisor of <math>a_n</math>.
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| − | As a consequence, every rational root of a [[monic polynomial]] with integral coefficients must be integral.
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| − | 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.
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| − | ==Problems==
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| − | ===Intermediate===
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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.
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