Rational Root Theorem

Pre-Calculus · Axiom Academy

Turn an infinite hunt for a polynomial's roots into a short, finite checklist — then test it down to the real answers. 1. From Infinitely Many to a Finite Few A degree-3 polynomial has 3 roots somewhere on (or off) the number line — but which numbers? You cannot test every real number; there are infinitely many. The theorem's payoff is that, if a root is rational, it can only be one of a short finite list you can write down in seconds and check one by one. Infinitely many candidates collapse to a handful 2. The Theorem: Where p and q Come From Write a rational root in lowest terms as . For a polynomial with integer coefficients , the numerator p must divide the constant term a_0 , and the denominator q must divide the leading coefficient a_n . So you only need the factors of those two numbers. Every rational root has this form; build the list by pairing each p with each q . Comes from the constant term a_0 . Use every positive and negative factor of a_0 . Comes from the leading coefficient a_n . Use its positive factors as denominators. Pair each p with each q , reduce, and keep both signs . Duplicates drop out. Then q = 1 and the list is just (factors of the constant) — all integers. 3. Worked Example: f(x) = 2x^3 - 3x^2 - 8x + 12 Read off a_0 = 12 and a_n = 2 . Factors of 12 give the numerators; factors of 2 give the denominators. Pairing them (both signs, reduced) is the full candidate list — then we test each to see which actually hit zero.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.