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

A divisão de polinômios usando divisão longa segue a mesma estrutura da divisão numérica.Neste método, o polinômio que está sendo dividido é o dividendo, aquele que o divide é o divisor, o resultado é o quociente e qualquer parte restante é o resto.A divisão começa escrevendo o dividendo e o divisor na forma padrão, com termos organizados em ordem decrescente de grau.O termo principal do dividendo é dividido pelo termo principal do divisor para determinar o primeiro termo do quociente.Este termo é então multiplicado por todo o divisor e o resultado é subtraído do dividendo.Depois disso, o próximo termo do dividendo é reduzido e o processo é repetido usando a expressão atualizada.Essas etapas continuam até que o grau da expressão restante seja menor que o grau do divisor, onde grau significa a maior potência da variável.O resultado final pode ser expresso como o quociente mais o resto sobre o divisor.Por exemplo, se a receita total e o custo por item de uma empresa forem modelados por polinômios, a divisão longa poderá ser usada para determinar a expressão que indica a receita por item.

divisão longa de polinômios

Long Division of Polynomials

A divis&#227;o de polin&#244;mios &#233; um processo alg&#233;brico essencial para simplificar express&#245;es e resolver equa&#231;&#245;es. Assim como a divis&#227;o num&#233;rica separa um n&#250;mero em quociente e resto, a divis&#227;o longa de polin&#244;mios particiona um polin&#244;mio em componentes mais simples; nesse contexto, o dividendo &#233; o polin&#244;mio que est&#225; sendo dividido, o divisor &#233; a express&#227;o que o divide e o resultado &#233; expresso em termos de um quociente e de um resto.A divis&#227;o come&#231;a com a organiza&#231;&#227;o do dividendo e do divisor na forma padr&#227;o &#8212; termos em pot&#234;ncias decrescentes da vari&#225;vel. O termo de maior grau do dividendo &#233; dividido pelo termo de maior grau do divisor para determinar o primeiro termo do quociente. Esse termo &#233; multiplicado por todo o divisor e o resultado &#233; subtra&#237;do do dividendo. Em seguida, o pr&#243;ximo termo &#233; trazido, e o processo se repete at&#233; que o resto tenha grau menor que o do divisor. O resultado &#233; ent&#227;o expresso 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 exemplo, ao 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>o primeiro termo do quociente &#233; 6x, seguido por &#8722;2, resultando no quociente final 6x &#8722; 2 e em um resto igual a 4. O resultado &#233; escrito 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>Esse processo pode ser generalizado: para polin&#244;mios P(x) e D(x), existem polin&#244;mios &#250;nicos Q(x) e R(x) tais 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>onde o grau de R(x) &#233; menor que o de D(x).

Divisão Longa de Polinômios

A divis&#227;o de polin&#244;mios &#233; um processo alg&#233;brico essencial para simplificar express&#245;es e resolver equa&#231;&#245;es. Assim como a ...

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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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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...

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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.

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Polynomial Division: Quotient and Remainder

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