Solucionador de equações cúbicas dedicado com raízes reais e complexas, etapas do método Cardano, gráficos cúbicos e exemplos resolvidos.
Calculadora do Teorema do Resto
Insira seus coeficientes polinomiais acima e clique em "Avalie f(c)" para ver os resultados.The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor (x − c), the remainder is exactly f(c). This means you can evaluate any polynomial at any point simply by performing synthetic division — the last number in the bottom row equals f(c).
The Factor Theorem is a direct corollary: if f(c) = 0, then (x − c) is a factor of f(x). These two theorems together provide a powerful bridge between evaluation and factoring. Instead of plugging in values manually (which involves large exponents), synthetic division gives the same answer with simpler arithmetic.
For cubic equations, the Remainder Theorem is especially useful for root verification. After finding candidate roots via the Rational Root Theorem, you can quickly confirm which ones are actual roots by checking whether f(c) = 0. This is faster and less error-prone than direct substitution, particularly for large coefficients.
Evaluate f(c) for any value c without computing large powers directly — synthetic division handles it cleanly.
After finding candidate roots, the Remainder Theorem instantly confirms which candidates are actual roots.
The theorem beautifully connects the concepts of division, evaluation, and factoring into one unified framework.
When dividing by (x+3), the evaluation point is c = −3, not c = 3. The theorem uses (x MINUS c).
The Remainder Theorem is not limited to cubics. It works for polynomials of any degree.
The remainder from division equals f(c). Don't confuse the quotient (a polynomial) with the remainder (a number).
| Theorem | f(x) ÷ (x−c) has remainder f(c) |
| Factor Test | f(c) = 0 means (x−c) is a factor |
| Method | Use synthetic division for efficiency |
| Works For | Polynomials of any degree |
| Key Benefit | Avoids computing large powers directly |
A divisão sintética fornece o quociente quadrático restante *e* o restante. Esta ferramenta ignora o quociente e fornece apenas o restante.
Sim! O restante<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">R</span>é literalmente o<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">sim</span>-coordenar no gráfico quando<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x = c</span>.
Parabéns! Você encontrou a raiz da equação por meio do Teorema do Fator.
O Teorema do Fator é um caso especial do Teorema do Resto. Se o resto f(c) = 0, então (x - c) é um fator do polinômio.
Sim, o Teorema do Resto funciona para polinômios de qualquer grau, não apenas para cúbicos. É uma ferramenta universal para avaliar valores polinomiais.
