Polinomios: División y Resto

Polynomial Remainder

How to Do It

To find the remainder of a polynomial division, follow these steps:

1. Divide the dividend polynomial by the divisor polynomial: Write the dividend polynomial (the polynomial being divided) on top of a line, and the divisor polynomial (the polynomial by which we are dividing) below it.
2. Divide the leading term of the dividend by the leading term of the divisor: Write the result below the line.
3. Multiply the result from step 2 by the divisor and subtract the product from the dividend: Bring down the next term of the dividend and repeat steps 2-3.
4. Continue these steps until the degree of the remainder is less than the degree of the divisor: The final result is the quotient and the remainder.

Facts to Do It

• The degree of the remainder is always less than the degree of the divisor: This is a fundamental property of polynomial long division.
• The remainder can be found using synthetic division: Synthetic division is a shortcut method for dividing polynomials. It is particularly useful when the divisor is a linear polynomial.
• The remainder theorem states that the remainder is equal to f(a) when dividing a polynomial f(x) by (x-a): This theorem can be used to quickly find the remainder of a polynomial division without performing the full division.
• The factor theorem states that (x-a) is a factor of f(x) if and only if f(a)=0: This theorem can be used to find the roots of a polynomial and to determine if a polynomial has a certain factor.

División de Polinomios

• Para encontrar el resto de una división de polinomios, se deben seguir los siguientes pasos:
  • Dividir el polinomio dividendo entre el polinomio divisor.
  • Dividir el término líder del dividendo entre el término líder del divisor y anotar el resultado debajo de la línea.
  • Multiplicar el resultado del paso 2 por el divisor y restar el producto del dividendo.
  • Repetir los pasos 2-3 hasta que el grado del resto sea menor que el grado del divisor.

Propiedades y Teoremas

Resto de la División de Polinomios

• El grado del resto es siempre menor que el grado del divisor.
• El resto se puede encontrar utilizando la división sintética, un método abreviado para dividir polinomios, especialmente útil cuando el divisor es un polinomio lineal.

Teorema del Resto

• Establece que el resto es igual a f(a) cuando se divide un polinomio f(x) entre (x-a).

Teorema del Factor

• Establece que (x-a) es un factor de f(x) si y solo si f(a) = 0, lo que permite encontrar raíces de un polinomio y determinar si un polinomio tiene un factor específico.