Dedikerad kubisk ekvationslösare med verkliga och komplexa rötter, Cardano-metodsteg, kubikgrafer och utarbetade exempel.
Rational Root Theorem Calculator
Ange dina polynomkoefficienter ovan och klicka på "Hitta rationella grundkandidater" för att se resultat.The Rational Root Theorem provides a systematic way to find all possible rational roots of a polynomial with integer coefficients. For a cubic ax³ + bx² + cx + d = 0, any rational root p/q must satisfy: p divides d (the constant term) and q divides a (the leading coefficient). This generates a finite list of candidates to test.
The theorem does NOT guarantee that rational roots exist — it only narrows the search space. You must test each candidate by substituting it into the polynomial (or using synthetic division). If f(p/q) = 0, you have found a root. Once one root is confirmed, synthetic division reduces the cubic to a quadratic, which the quadratic formula solves completely.
The power of this theorem lies in its efficiency: instead of guessing randomly, you have a guaranteed finite list. For example, if a = 2 and d = 12, the candidates are ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2 — at most 16 values to check. This structured approach is the standard first step in polynomial solving before resorting to Cardano's method.
The theorem is always the first tool applied when solving cubics with integer coefficients — before Cardano or numerical methods.
Most algebra and precalculus exams feature problems solvable by the Rational Root Theorem, making it essential test knowledge.
Computer algebra systems use the Rational Root Theorem as the initial step in their polynomial factorization algorithms.
Every candidate ±p/q has both positive and negative versions. Testing only positives misses negative roots.
Candidates like 2/4 and 1/2 are the same root. Reduce fractions to avoid redundant testing.
p divides the CONSTANT term d, and q divides the LEADING coefficient a. Swapping them generates wrong candidates.
| Rule | p divides d, q divides a |
| Candidate Form | ±p/q (all combinations) |
| Testing Method | Substitute or synthetic division |
| Limitation | Only finds rational roots, not irrational |
| After Finding Root | Use synthetic division to reduce degree |
Nej, det ger dig bara en "shortlist" med *kandidater*. Du måste testa dem för att se vilken som är lika med noll.
Det betyder att ekvationen har irrationella rötter (stökiga decimaler eller kvadratrötter) och måste lösas med avancerade formler som Cardanos.
Nej, satsen bygger otroligt nog bara på de ledande och konstanta termerna.
Antalet kandidater beror på hur många faktorer ledande koefficient och konstant term har. Större antal med många faktorer ger längre kandidatlistor.
Nej. Den rationella rotsatsen identifierar endast potentiella rationella (heltal eller bråk) rötter. Irrationella rötter som √2 kräver andra metoder.
