The Rational Zeros Theorem: Theory and Step-by-Step Factoring
The Rational Root Theorem (or Rational Zeros Theorem) is a foundational algebraic principle that narrows down the infinitely many potential roots of a polynomial with integer coefficients to a finite candidate list.
Theorem Formulation
Let $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ be a polynomial with integer coefficients, where $a_n \ne 0$ and $a_0 \ne 0$.
If $x = \frac{p}{q}$ is a rational root in lowest terms (so $\gcd(p, q) = 1$), then:
- 1The numerator $p$ is an integer factor of the constant term $a_0$.
- 2The denominator $q$ is an integer factor of the leading coefficient $a_n$.
Step-by-Step Discovery Protocol
- Step 1: Find all positive integer divisors of $|a_0|$ to form the list of possible values for $p$.
- Step 2: Find all positive integer divisors of $|a_n|$ to form the list of possible values for $q$.
- Step 3: Form all possible quotients $\pm \frac{p}{q}$, reduce fractions to lowest terms, and remove duplicates.
- Step 4: Test each candidate using synthetic division or polynomial evaluation $P(p/q) = 0$.
Example Application: $2x^3 + 3x^2 - 8x - 6 = 0$
| Coefficient | Factors | Possible Values |
|---|---|---|
| Constant $a_0 = -6$ | Divisors of 6 | $p \in \{1, 2, 3, 6\}$ |
| Leading $a_n = 2$ | Divisors of 2 | $q \in \{1, 2\}$ |
| Ratios $\pm p/q$ | Combined | $\pm 1, \pm 1/2, \pm 2, \pm 3, \pm 3/2, \pm 6$ |