Rational root theorem

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In algebra, the rational root theorem states that given an integer polynomial $P(x)$ with leading coefficient $a_n$ and constant term $a_0$, if $P(x)$ has a rational root $r = \frac pq$ in lowest terms, then $p|a_0$ and $q|a_n$.

This theorem is most often used to guess the roots of polynomials.

Proof

Let $\frac{p}{q}$ be a rational root of $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0$, where every $a_r$ is an integer; we wish to show that $p|a_0$ and $q|a_n$. Since $\frac{p}{q}$ is a root of $P(x)$, $$0 = a_n \left(\frac{p}{q}\right)^n + a_{n-1} \left(\frac{p}{q}\right)^{n-1} + \cdots + a_1 \left(\frac{p}{q}\right) + a_0.$$ Multiplying by $q^n$ yields $$0 = a_n p^n + a_{n-1} p^{n-1} q + \cdots + a_1 p * q^{n-1} + a_0 q^n.$$ Using modular arithmetic modulo $p$, we have $a_0 q^n \equiv 0\pmod p$, which implies that $p | a_0 q^n$. Because we've defined $p$ and $q$ to be relatively prime, $\gcd(q^n, p) = 1$, which implies $p | a_0$ by Euclid's lemma. Via similar logic in modulo $q$, $q|a_n$, as required. $\square$

Examples

Here are some problems that are cracked by the rational root theorem.

Example 1

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

Solution: The polynomial has leading coefficient $1$ and constant term $3 \cdot 19$, so the rational root theorem guarantees that the only possible rational roots are $-57$, $-19$, $-3$, $-1$, $1$, $3$, $19$, and $57$. After testing every number, we find that none of these are roots of the polynomial; thus, the polynomial has no rational roots. $\square$

Example 2

Factor the polynomial $x^3-5x^2+2x+8$.

Solution: After testing the divisors of 8, we find that it has roots $-1$, $2$, and $4$. Then because it has leading coefficient $1$, the factor theorem tells us that it has the factorization $(x-4)(x-2)(x+1)$. $\square$

Example 3

Using the rational root theorem, prove that $\sqrt{2}$ is irrational.

Solution: The polynomial $x^2 - 2$ has roots $-\sqrt{2}$ and $\sqrt{2}$. The rational root theorem garuntees that the only possible rational roots of this polynomial are $-2, -1, 1$, and $2$. Testing these, we find that none are roots of the polynomial, and so it has no rational roots. Then because $\sqrt{2}$ is a root of the polynomial, it cannot be a rational number. $\square$