Name: polynomial division: quotient and remainder
Uploaded: 2025-11-26T05:43:58.000Z
Duration: 1 min 26 s
Channel: JoVE

La división de polinomios usando la división larga sigue la misma estructura que la división numérica.En este método, el polinomio que se divide es el dividendo, el que lo divide es el divisor, el resultado es el cociente y cualquier parte restante es el resto.La división comienza escribiendo el dividendo y el divisor en forma estándar, con términos dispuestos en orden descendente de grado.El término principal del dividendo se divide por el término principal del divisor para determinar el primer término del cociente.Este término se multiplica por todo el divisor y el resultado se resta del dividendo.Después de esto, se reduce el siguiente término del dividendo y el proceso se repite utilizando la expresión actualizada.Estos pasos continúan hasta que el grado de la expresión restante es menor que el grado del divisor, donde grado significa la potencia más alta de la variable.El resultado final se puede expresar como el cociente más el resto sobre el divisor.Por ejemplo, si los ingresos totales y el costo por artículo de una empresa se modelan mediante polinomios, se puede usar la división larga para determinar la expresión que indica los ingresos por artículo.

división larga de polinomios

Long Division of Polynomials

La divisi&#243;n de polinomios es un proceso algebraico esencial para simplificar expresiones y resolver ecuaciones. As&#237; como la divisi&#243;n num&#233;rica separa un n&#250;mero en cociente y residuo, la divisi&#243;n larga de polinomios descompone un polinomio en componentes m&#225;s simples; en este contexto, el dividendo es el polinomio que se divide, el divisor es la expresi&#243;n que lo divide y el resultado se expresa en t&#233;rminos de cociente y residuo.La divisi&#243;n comienza ordenando el dividendo y el divisor en forma can&#243;nica (t&#233;rminos en potencias decrecientes de la variable). El primer t&#233;rmino del dividendo se divide por el t&#233;rmino principal del divisor para determinar el primer t&#233;rmino del cociente. Este t&#233;rmino se multiplica por el divisor completo y el resultado se sustrae del dividendo. A continuaci&#243;n, se baja el siguiente t&#233;rmino del dividendo y el proceso se repite hasta que el residuo tenga un grado menor que el del divisor. El resultado se expresa entonces como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40055/40055_Equation_1_1__editor_40055.svg' alt='40055_Equation_1_1_.svg' /></figure>Por ejemplo, al dividir:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40055/40055_Equation_2_1__editor_40055.svg' alt='40055_Equation_2_1_.svg' /></figure>el primer t&#233;rmino del cociente es 6x, seguido de &#8722;2, lo que da como resultado el cociente final 6x &#8722; 2 y un residuo de 4. El resultado se escribe como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40055/40055_Equation_3_1__editor_40055.svg' alt='40055_Equation_3_1_.svg' /></figure>Este proceso puede generalizarse: para los polinomios P(x) y D(x), existen polinomios &#250;nicos Q(x) y R(x) tales que<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40055/40055_Equation_4_2__editor_40055.svg' alt='40055_Equation_4_2_.svg' /></figure>donde el grado de R(x) es menor que el grado de D(x).

División larga de polinomios

La divisi&#243;n de polinomios es un proceso algebraico esencial para simplificar expresiones y resolver ecuaciones. As&#237; como la divisi&#243;n ...

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polynomial division: quotient and remainder

Polynomial division breaks a polynomial into a quotient and a remainder. It works like numerical division, but the dividend is a polynomial and the divisor is an algebraic expression. The result shows how much of the divisor fits into the dividend and what is left over.

The first step is to write both polynomials in standard form, with terms in descending powers of the variable. Then divide the leading term of the dividend by the leading term of the divisor. That first result becomes the first term of the quotient.

Next, multiply that term by the entire divisor and subtract the result from the dividend. Bring down the next term and repeat the same steps. Continue until the remainder has a degree that is less than the degree of the divisor.

The final answer is written as a quotient plus a remainder. For example, dividing a polynomial can produce a quotient of 6x - 2 with a remainder of 4. In general, for polynomials P(x) and D(x), there are unique polynomials Q(x) and R(x) such that P(x) = D(x)Q(x) + R(x), and the degree of R(x) is less than the degree of D(x).

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Learn polynomial long division by finding the quotient and remainder step by step. See how to use standard form, subtract, and stop when the remainder is smaller.

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The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Polynomial Division: Quotient and Remainder

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