Solucionador de ecuaciones cúbicas dedicado con raíces reales y complejas, pasos del método Cardano, gráficas cúbicas y ejemplos resueltos.
Calculadora de División Sintética
Ingrese los coeficientes de su polinomio arriba y haga clic en "Realizar división sintética" para ver los resultados.Synthetic division is a streamlined algorithm for dividing a polynomial by a linear factor of the form (x − c). Instead of writing out full polynomial long division with variables and exponents, synthetic division uses only the coefficients arranged in a compact table, making it dramatically faster and less error-prone.
The process works by “cascading” multiplications and additions: bring down the leading coefficient, multiply by c, add to the next coefficient, multiply by c again, and repeat. The final number in the row is the remainder. If the remainder is zero, then c is a root and (x−c) is a factor. The remaining numbers form the quotient polynomial of one degree less.
Synthetic division serves dual purposes: division (finding the quotient when you know a factor) and evaluation (the remainder equals f(c) by the Remainder Theorem). This duality makes it an essential tool in the systematic root-finding process for cubics and higher-degree polynomials.
Quickly test whether a candidate value is a root by checking if the remainder is zero — much faster than substitution.
After finding one root, synthetic division reduces the cubic to a quadratic, enabling immediate use of the quadratic formula.
Students can verify factoring homework in seconds using the compact synthetic division format.
When dividing by (x + 3), the divisor value is −3, not +3. The sign convention often trips students up.
If a power is missing (e.g., no x² term), you MUST insert a 0 coefficient for that position.
Synthetic division only works for linear divisors (x − c). For quadratic or higher divisors, use polynomial long division.
| Divisor Form | (x − c) only |
| Remainder = 0 | c is a root, (x−c) is a factor |
| Remainder ≠ 0 | Remainder equals f(c) |
| Speed | ~3× faster than polynomial long division |
| Output | Quotient polynomial + remainder |
No, la división sintética estándar sólo funciona perfectamente para dividir por binomios lineales de la forma<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x-c</span>.
Sí. Si tu cúbica es<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x³ - 7x + 6</span>, debes tratar el<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span>coeficiente como 0. La herramienta maneja automáticamente los ceros ingresados.
Entonces el número que probaste no es una raíz, pero el resto representa matemáticamente la evaluación.<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">f(r)</span>.
La división sintética es un atajo que elimina todas las variables y solo funciona con coeficientes. Es más rápido y menos propenso a errores para divisores lineales, pero la división larga maneja cualquier grado de divisor.
¡Sí! Si el resto es cero después de la división sintética, el número que probaste es de hecho una raíz del polinomio.
