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

Die synthetische Division ist eine vereinfachte Methode, um Polynome nur anhand ihrer Koeffizienten zu dividieren. Es wird verwendet, wenn der Divisor ein lineares Binom der Form x - c ist, wobei c eine reelle Zahl ist.Der Prozess beginnt damit, die Koeffizienten der Dividende in absteigender Gradreihenfolge zu schreiben und Nullen für alle fehlenden Grade zwischen dem führenden Term und der Konstanten einzufügen. Die Konstante c aus dem Divisor wird auf der linken Seite des Divisionsaufbaus platziert.Der erste Koeffizient wird unter der Linie geschrieben. Es wird dann mit der Konstanten multipliziert, und das Ergebnis wird zum nächsten Koeffizienten addiert. Dieser Multiplikations- und Addierzyklus wird in der gesamten Zeile fortgesetzt.Der letzte Wert in der Zeile stellt den Rest dar, während die vorhergehenden Werte die Koeffizienten des Quotienten bilden – ein Polynom mit einem Grad eins kleiner als der des Dividends. Das Endergebnis kann ausgedrückt werden als Dividende gleich Divisor mal Quotient plus Rest.In der Wirtschaft können Polynome den Gewinn basierend auf dem Produktionsniveau modellieren. Um den Gewinn auf einem bestimmten Niveau zu ermitteln, wird das Polynom durch ein Binom x minus dieses Niveaus dividiert.Die Verwendung der synthetischen Divisionsmethode für diese Division ergibt den Rest, der den Gewinn auf dieser Ebene darstellt.

synthetische division von polynomen

Synthetic Disvision of Polynomials

Die synthetische Division ist ein effizienter algorithmischer Ansatz zur Division eines Polynoms durch ein lineares Binom der Form x &#8722; c, wobei c eine reelle Zahl ist. Diese Methode ist aufgrund ihres schlanken Ablaufs hilfreich, da sie die aufw&#228;ndigeren Schritte der herk&#246;mmlichen schriftlichen Polynomdivision vermeidet. Sie vereinfacht die Rechnung und dient als praktisches Werkzeug zur Auswertung von Polynomen und zur Bestimmung ihrer Faktoren.
Zur Durchf&#252;hrung der synthetischen Division werden zun&#228;chst die Koeffizienten des Dividendenpolynoms in absteigender Potenzordnung notiert. Fehlende Terme werden zur Ausrichtung mit dem Koeffizienten 0 erg&#228;nzt. Die Konstante c aus dem Divisor x &#8722; c wird links in das Schema eingetragen. Der Vorgang beginnt mit dem f&#252;hrenden Koeffizienten, der unver&#228;ndert heruntergezogen wird. Dieser Wert wird mit c multipliziert; das Ergebnis wird zum n&#228;chsten Koeffizienten addiert. Dieses abwechselnde Multiplizieren und Addieren wird fortgesetzt, bis alle Koeffizienten verarbeitet sind.
Die in der letzten Zeile erhaltenen Zahlen stellen die Koeffizienten des Quotientenpolynoms dar; der letzte Eintrag ist der Rest. Beispielsweise ergibt die Division von 2x^3 &#8722; 6x^2 + 2x &#8722; 4 durch x &#8722; 2 mittels synthetischer Division die Koeffizienten eines quadratischen Polynoms und damit den Quotienten 2x^2 &#8722; 2x &#8722; 2 sowie einen Rest von 0 &#8212; dies zeigt, dass x &#8722; 2 ein Faktor des Polynoms ist.
Dieses Ergebnis veranschaulicht unmittelbar den Restsatz, der besagt, dass der Rest bei der Division eines Polynoms f(x) durch x &#8722; c gleich f(c) ist. Ist dieser Rest 0, best&#228;tigt der Faktorsatz, dass x &#8722; c ein Faktor von f(x) ist.

Synthetische Division von Polynomen

Die synthetische Division ist ein effizienter algorithmischer Ansatz zur Division eines Polynoms durch ein lineares Binom der Form x &#8722; c, wobei 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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Learn how synthetic division divides polynomials by x - c, finds the quotient and remainder, and shows when a binomial is a factor.

Using Synthetic Division to Find Factors | JoVE K12

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Using Synthetic Division to Find Factors

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