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

La division de polynômes à l’aide de la division longue suit la même structure que la division numérique.Dans cette méthode, le polynôme divisé est le dividende, celui qui le divise est le diviseur, le résultat est le quotient, et toute partie restante est le reste.La division commence par l’écriture du dividende et du diviseur sous forme standard, avec des termes disposés par ordre décroissant de degré.Le terme principal du dividende est divisé par le terme principal du diviseur pour déterminer le premier terme du quotient.Ce terme est ensuite multiplié par le diviseur entier, et le résultat est soustrait du dividende.Après cela, le terme suivant du dividende est abaissé et le processus est répété à l’aide de l’expression mise à jour.Ces étapes se poursuivent jusqu’à ce que le degré de l’expression restante soit inférieur au degré du diviseur, où degré signifie la puissance la plus élevée de la variable.Le résultat final peut être exprimé comme le quotient plus le reste sur le diviseur.Par exemple, si le revenu total et le coût par article d’une entreprise sont modélisés par des polynômes, la division longue peut être utilisée pour déterminer l’expression indiquant le revenu par article.

division euclidienne des polynômes

Long Division of Polynomials

La division des polyn&#244;mes est un proc&#233;d&#233; alg&#233;brique fondamental permettant de simplifier des expressions et de r&#233;soudre des &#233;quations. De la m&#234;me mani&#232;re que la division num&#233;rique s&#233;pare un nombre en quotient et en reste, la division euclidienne des polyn&#244;mes d&#233;compose un polyn&#244;me en &#233;l&#233;ments plus simples. Dans ce cadre, le dividende correspond au polyn&#244;me &#224; diviser, le diviseur au polyn&#244;me diviseur, et le r&#233;sultat s&#8217;exprime sous la forme d&#8217;un quotient et d&#8217;un reste.La division d&#233;bute par l&#8217;&#233;criture du dividende et du diviseur selon l&#8217;ordre d&#233;croissant des degr&#233;s. Le terme de plus haut degr&#233; du dividende est alors divis&#233; par le terme de plus haut degr&#233; du diviseur pour d&#233;terminer le premier terme du quotient. Ce terme est ensuite multipli&#233; par le diviseur tout entier, puis le produit obtenu est soustrait du dividende. Le terme suivant du dividende est ensuite abaiss&#233;, et le processus se r&#233;p&#232;te jusqu&#8217;&#224; ce que le reste ait un degr&#233; strictement inf&#233;rieur &#224; celui du diviseur. Le r&#233;sultat s&#8217;exprime alors comme suit :<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>Par exemple, en effectuant la division&#160;:<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>le premier terme du quotient est 6x, suivi de &#8722;2, ce qui conduit au quotient final 6x &#8722; 2 et &#224; un reste &#233;gal &#224; 4. Le r&#233;sultat s&#8217;&#233;crit ainsi&#160;:<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>Ce proc&#233;d&#233; se g&#233;n&#233;ralise : pour deux polyn&#244;mes P(x) et D(x), il existe des polyn&#244;mes uniques Q(x) et R(x) tels 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>o&#249; le degr&#233; de R(x) est strictement inf&#233;rieur &#224; celui de D(x).

Division euclidienne des polynômes

La division des polyn&#244;mes est un proc&#233;d&#233; alg&#233;brique fondamental permettant de simplifier des expressions et de r&#233;soudre des ...

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Math Fundamentals

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Polynomial and Rational Functions

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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).

Polynomial Division: Quotient and Remainder

polynomial-division-quotient-and-remainder

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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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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 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.

math fundamentals

Polynomial Division: Quotient and Remainder

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