Name: using synthetic division to find factors
Uploaded: 2025-11-26T05:43:56.000Z
Duration: 1 min 28 s
Channel: JoVE

La division synthétique est une méthode simplifiée pour diviser les polynômes en utilisant uniquement leurs coefficients. Il est utilisé lorsque le diviseur est un binôme linéaire de la forme x - c, où c est un nombre réel.Le processus commence par l’écriture des coefficients du dividende par ordre décroissant de degré, en insérant des zéros pour tous les degrés manquants entre le terme principal et la constante. La constante c du diviseur est placée sur le côté gauche de la configuration de la division.Le premier coefficient est écrit en dessous de la ligne. Il est ensuite multiplié par la constante, et le résultat est ajouté au coefficient suivant. Ce cycle de multiplication et d’addition se poursuit sur toute la rangée.La dernière valeur de la ligne représente le reste, tandis que les valeurs précédentes forment les coefficients du quotient, un polynôme dont le degré est inférieur d’un à celui du dividende. Le résultat final peut être exprimé comme le dividende est égal au diviseur multiplié par le quotient plus le reste.En affaires, les polynômes peuvent modéliser le profit en fonction du niveau de production. Pour trouver le profit à un niveau spécifique, le polynôme est divisé par un binôme x moins ce niveau.L’utilisation de la méthode de division synthétique pour cette division donne le reste, qui représente le profit à ce niveau.

division synthétique des polynômes

Synthetic Disvision of Polynomials

La division synth&#233;tique est une approche algorithmique efficace pour diviser un polyn&#244;me par un bin&#244;me de degr&#233; 1 de la forme x - c, o&#249; c d&#233;signe un r&#233;el. Cette m&#233;thode est avantageuse en raison d&#8217;un proc&#233;d&#233; simplifi&#233; qui &#233;vite les &#233;tapes plus lourdes de la division euclidienne des polyn&#244;mes. Elle all&#232;ge les calculs et constitue un outil pratique pour &#233;valuer des polyn&#244;mes et identifier leurs facteurs.
Pour effectuer une division synth&#233;tique, on commence par &#233;num&#233;rer les coefficients du polyn&#244;me dividende par ordre d&#233;croissant des puissances. Si des termes manquent, les coefficients manquants sont compl&#233;t&#233;s par des z&#233;ros afin de pr&#233;server l&#8217;alignement. La constante c du diviseur x - c est plac&#233;e &#224; gauche du tableau de division synth&#233;tique. Le processus d&#233;bute par le coefficient dominant, abaiss&#233; tel quel. Cette valeur est multipli&#233;e par c, puis le r&#233;sultat est ajout&#233; au coefficient suivant. Cette alternance de multiplications et d&#8217;additions se poursuit jusqu&#8217;&#224; ce que tous les coefficients aient &#233;t&#233; trait&#233;s.
Les nombres obtenus sur la derni&#232;re ligne repr&#233;sentent les coefficients du polyn&#244;me quotient, &#224; l&#8217;exception de la derni&#232;re valeur, qui est le reste. Par exemple, la division de 2x^3 - 6x^2 + 2x - 4 par x - 2 &#224; l&#8217;aide de la division synth&#233;tique fournit les coefficients d&#8217;un polyn&#244;me du second degr&#233;, ce qui donne le quotient 2x^2 - 2x - 2 et un reste nul, indiquant que x - 2 est un facteur.
Ce r&#233;sultat illustre directement le th&#233;or&#232;me du reste, qui &#233;nonce que le reste de la division d&#8217;un polyn&#244;me f(x) par x - c est f(c). Si ce reste est nul, le th&#233;or&#232;me du facteur confirme que x - c est un facteur de f(x).

Division synthétique des polynômes

La division synth&#233;tique est une approche algorithmique efficace pour diviser un polyn&#244;me par un bin&#244;me de degr&#233; 1 de la forme x - c, ...

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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 process begins by bringing down the first coefficient unchanged. That number is then multiplied by c and added to the next coefficient. The same multiply-and-add pattern continues until every coefficient has been used.

The final row gives the quotient coefficients, except for the last number. That last number is the remainder. For example, dividing 2x^3 - 6x^2 + 2x - 4 by x - 2 gives the quotient 2x^2 - 2x - 2 and a remainder of zero.

A remainder of zero shows that x - 2 is a factor of the polynomial. This connects synthetic division to the Remainder Theorem, which says the remainder from dividing f(x) by x - c is f(c). If that remainder is zero, the Factor Theorem confirms that x - c is a factor of f(x).

Using Synthetic Division to Find Factors

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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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Learn how synthetic division divides polynomials by x - c, finds the quotient and remainder, and shows when a binomial is a factor.

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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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Using Synthetic Division to Find Factors

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