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

La divisione dei polinomi utilizzando la divisione lunga segue la stessa struttura della divisione numerica.In questo metodo, il polinomio diviso è il dividendo, quello che lo divide è il divisore, il risultato è il quoziente e l'eventuale parte rimanente è il resto.La divisione inizia con la scrittura del dividendo e del divisore in forma standard, con termini disposti in ordine decrescente di grado.La durata iniziale del dividendo è divisa per la durata iniziale del divisore per determinare la prima durata del quoziente.Questo termine viene quindi moltiplicato per l'intero divisore e il risultato viene sottratto dal dividendo.Successivamente, il termine successivo del dividendo viene abbassato e il processo viene ripetuto utilizzando l'espressione aggiornata.Questi passaggi continuano fino a quando il grado dell'espressione rimanente è inferiore al grado del divisore, dove grado indica la potenza più alta della variabile.Il risultato finale può essere espresso come quoziente più il resto sul divisore.Ad esempio, se il fatturato totale e il costo per articolo di un'azienda sono modellati da polinomi, la divisione lunga può essere utilizzata per determinare l'espressione che indica il ricavo per articolo.

divisione lunga tra polinomi

La divisione polinomiale &#232; un procedimento algebrico essenziale per semplificare espressioni e risolvere equazioni. Analogamente alla divisione numerica, che separa un numero in quoziente e resto, la divisione lunga tra polinomi suddivide un polinomio in componenti pi&#249; semplici; in questo contesto, il dividendo &#232; il polinomio da dividere, il divisore &#232; l'espressione che lo divide e il risultato si esprime in termini di quoziente e resto.La divisione inizia disponendo il dividendo e il divisore in forma standard, cio&#232; con i termini disposti in ordine decrescente di grado della variabile. Il termine di grado massimo del dividendo viene diviso per il termine di grado massimo del divisore per determinare il primo termine del quoziente. Questo termine viene moltiplicato per l'intero divisore e il risultato viene sottratto dal dividendo. Il termine successivo del dividendo viene riportato e il processo si ripete finch&#233; il grado del resto non &#232; inferiore a quello del divisore. Il risultato si esprime quindi come:<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>Ad esempio, dividendo:<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>il primo termine del quoziente &#232; 6x, seguito da &#8722;2; si ottiene cos&#236; il quoziente finale 6x &#8722; 2 e un resto pari a 4. Il risultato si scrive come:<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>Questo processo pu&#242; essere generalizzato: per i polinomi P(x) e D(x), esistono polinomi unici Q(x) e R(x) tali che:<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>dove il grado di R(x) &#232; strettamente inferiore a quello di D(x).

Divisione lunga tra polinomi

La divisione polinomiale &#232; un procedimento algebrico essenziale per semplificare espressioni e risolvere equazioni. Analogamente alla divisione ...

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Math Fundamentals

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Polynomial and Rational Functions

Funzioni polinomiali e razionali

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

Long Division of Polynomials

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).

Polynomial Division: Quotient and Remainder

polynomial-division-quotient-and-remainder

quadratic models

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long division of polynomials

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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Using Synthetic Division to Find Factors

Synthetic division is a fast way to divide a polynomial by a linear binomial of the form x - c, where c is a real number. It gives a shorter method than polynomial long division and is useful for checking values and finding factors. The setup starts with the coefficients of the polynomial written in descending order of powers. If a term is missing, a zero must be used to keep the columns lined up. The constant c from the divisor x - c is placed to the left of the synthetic division table. The...

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key terms and concepts

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.

Polynomial Division: Quotient and Remainder | JoVE K12

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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