Algebra theory - Rational Root Theorem

Rational Root Theorem (Formal Statement):

Let

$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$

be a polynomial with integer coefficients, where $a_0, a_1, \dots, a_n \in \mathbb{Z}$ and $a_n \ne 0$.

If $\frac{p}{q}$ is a rational root of $f(x)$, expressed in lowest terms (i.e., $\gcd(p, q) = 1$), then:

• $p$ divides the constant term $a_0$, and

• $q$ divides the leading coefficient $a_n$.

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In other words:

If $\frac{p}{q} \in \mathbb{Q}$ is a root of $f(x)$ and $\gcd(p, q) = 1$, then:

$$p \mid a_0 \quad \text{and} \quad q \mid a_n$$

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