Recipe

Solve a polynomial equation using the Rational Zero Theorem

Use when: you're given a polynomial equation of degree 3 or higher, with integer coefficients, that doesn't factor by inspection

Goal: every real zero of the polynomial, found exactly, with no rounding

Taught in The Rational Zero Theorem

1. 1

   List every divisor of the constant term a_0 (with both signs) as possible values of p, and every divisor of the leading coefficient a_n as possible values of q. Every ratio p/q is one candidate.

   Example

   For f(x) = 2x^3 - 3x^2 - 11x + 6: p divides 6 (\pm1, \pm2, \pm3, \pm6), q divides 2 (1, 2), giving the candidate list \pm1, \pm2, \pm3, \pm6, \pm\tfrac12, \pm\tfrac32.

   Why: The Rational Zero Theorem guarantees any rational zero of f must appear on this list — nowhere else.

2. 2

   Pick a candidate, integers first since they're cheapest to test, and run synthetic division by it against the current polynomial. A remainder of 0 means you found a zero; anything else means move to the next candidate.

   Example

   Testing x = 3 against the coefficients 2, -3, -11, 6 finishes with remainder 0, so 3 is a zero of f.

   • Remainder is 0 — continue to Step 3 and divide the factor out.
   • Remainder is nonzero — discard this candidate and test the next one from Step 1's list.

   Why: A zero remainder is exactly the Factor Theorem's test — it means (x - c) is a genuine factor, not just a lucky evaluation.

3. 3

   The bottom row of that synthetic division is the quotient, one degree lower than what you just tested. Set the original polynomial aside and work only with this smaller quotient from here on.

   Example

   The bottom row 2, 3, -2 gives the quotient 2x^2 + 3x - 2: the search has shrunk from a cubic down to a quadratic in one step.

   Why: Every zero of the quotient is also a zero of the original polynomial, and a smaller polynomial means cheaper, shorter synthetic-division rows on the next pass.

4. 4

   As long as the quotient's degree is 3 or higher, return to Step 2 and test the same candidate list against it (a candidate can repeat if a zero has multiplicity greater than 1). The moment the quotient is a quadratic, stop dividing: factor it directly, or fall back to the quadratic formula if it doesn't factor by inspection.

   Example

   Here the quotient 2x^2 + 3x - 2 is already a quadratic. It factors as (2x - 1)(x + 2), giving the last two zeros x = \tfrac12 and x = -2.

   • The quotient's degree is 3 or higher — return to Step 2 on this smaller quotient.
   • The quotient is a quadratic — factor it, or use the quadratic formula, to find its last two zeros exactly, real or complex.

   Why: The quadratic formula always finishes a quadratic exactly, so there's no reason to keep hunting rational candidates once the degree drops to 2.