Rational Root Theorem
The Rational Root Theorem is a Grade 11 algebra tool in Big Ideas Math for finding potential rational roots of a polynomial equation. If a polynomial aₙxⁿ + ... + a₀ has integer coefficients, then any rational root p/q (in lowest terms) must have p as a factor of the constant term a₀ and q as a factor of the leading coefficient aₙ. For example, for 2x³ − 3x² − 11x + 6, possible rational roots are ±1, ±2, ±3, ±6, ±1/2, ±3/2. These candidates are tested by substitution or synthetic division. The theorem narrows the search before applying factoring or numerical methods.
Key Concepts
For a polynomial $P(x) = a n x^n + a {n 1} x^{n 1} + \cdots + a 1 x + a 0$ with integer coefficients, any rational root $\frac{p}{q}$ (in lowest terms) must satisfy: $p$ divides the constant term $a 0$ $q$ divides the leading coefficient $a n$.
Common Questions
What does the Rational Root Theorem state?
If a polynomial with integer coefficients has a rational root p/q (in lowest terms), then p must be a factor of the constant term and q must be a factor of the leading coefficient.
How do you list possible rational roots using the Rational Root Theorem?
List all factors of the constant term (±p values) and all factors of the leading coefficient (±q values), then form all combinations p/q.
For 2x³ − 3x² − 11x + 6, what are the possible rational roots?
Factors of 6: ±1, ±2, ±3, ±6. Factors of 2: ±1, ±2. Possible roots: ±1, ±2, ±3, ±6, ±1/2, ±3/2.
How do you test whether a possible rational root is actually a root?
Substitute the value into the polynomial (if result is 0, it's a root) or use synthetic division (if remainder is 0, it's a root).
What do you do after finding one rational root?
Use synthetic or polynomial division to factor out (x − root) from the polynomial, reducing its degree. Repeat on the quotient to find additional roots.
Does the Rational Root Theorem find irrational or complex roots?
No—it only generates candidates for rational roots. Irrational and complex roots must be found using the quadratic formula or numerical methods after rational roots are removed.