Field theory and polynomials Polynomials in one variable Theorem

Rational root theorem

Statement

Let $f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb{Z}[x]$ with $n \ge 1$ and $a_n \neq 0$ . Let $r = p/q$ be a rational root of $f$ , with $p, q \in \mathbb{Z}$ , $q > 0$ and $\gcd(p, q) = 1$ . Then $p$ divides $a_0$ and $q$ divides $a_n$ .

Remarks

The proof: multiply $f(p/q) = 0$ by $q^n$ . Then $a_n p^n = -q(\dots)$ and $a_0 q^n = -p(\dots)$ , and $\gcd(p, q) = 1$ gives the two divisibilities. For example, the only possible rational roots of $2x^3 - 3x + 1$ are $\pm 1$ and $\pm \tfrac{1}{2}$ . Evaluation shows that $1$ is a root. The same statement holds over every unique factorisation domain and its field of fractions. For a monic $f$ , every rational root is an integer.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Group concept
Ring definition
Polynomial ring definition
Polynomial evaluation definition
Root of a polynomial definition
Rational root theorem statement

Where this leads

2 pages build on this one.

Tag	`0819CQ1V8H`
URI	`m2://0819CQ1V8H`
Aliases	`rational-root-test`
Kind	statement, theorem
Writing	W1 stated
MSC	12E05
Status	This copy of the site has no build data. Run `m2 status 0819CQ1V8H`.
Uses	Root of a polynomial
Same as	Wikidata Q180345 Wikipedia Rational root theorem

Rational root theorem

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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