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

Synthetic division is a simplified method for dividing polynomials using only their coefficients. It is used when the divisor is a linear binomial of the form x - c, where c is a real number.The process begins by writing the dividend's coefficients in descending order of degree, inserting zeros for any missing degrees between the leading term and the constant. The constant c from the divisor is placed on the left side of the division setup.The first coefficient is written below the line. It is then multiplied by the constant, and the result is added to the next coefficient. This multiply-and-add cycle continues across the row.The last value in the row represents the remainder, while the preceding values form the coefficients of the quotient&#8212;a polynomial with a degree one less than that of the dividend. The final result can be expressed as dividend equals divisor times quotient plus remainder.In business, polynomials can model profit based on production level. To find the profit at a specific level, the polynomial is divided by a binomial x minus that level.Using the synthetic division method for this division gives the remainder, which represents the profit at that level.

synthetische deling van polynomen

Synthetic Disvision of Polynomials

Synthetische deling is een effici&#235;nte, algoritmische methode om een polynoom te delen door een lineair binomium van de vorm x &#8722; c, waarbij c een re&#235;el getal is. Deze aanpak is nuttig door het gestroomlijnde proces, waardoor de omslachtigere stappen van de traditionele lange deling van polynomen worden vermeden. Deze methode vereenvoudigt de berekening en is een praktisch hulpmiddel om polynomen te evalueren en hun factoren te identificeren.
Om synthetische deling uit te voeren, noteert men eerst de co&#235;ffici&#235;nten van het polynomiale deeltal in dalende volgorde van machten. Ontbreekt een term, dan wordt de bijbehorende co&#235;ffici&#235;nt als 0 genoteerd om de uitlijning te behouden. De constante c uit de deler x &#8722; c wordt links in het schema geplaatst. Het proces start met de leidende co&#235;ffici&#235;nt, die ongewijzigd naar beneden wordt overgenomen. Deze waarde wordt met c vermenigvuldigd en het resultaat wordt bij de volgende co&#235;ffici&#235;nt opgeteld. Deze cyclus van vermenigvuldigen en optellen gaat door totdat alle co&#235;ffici&#235;nten zijn verwerkt.
De getallen in de laatste rij vormen de co&#235;ffici&#235;nten van het quoti&#235;ntpolynoom; de laatste waarde is de rest. Zo levert het delen van 2x^3 &#8722; 6x^2 + 2x &#8722; 4 door x &#8722; 2 met synthetische deling de co&#235;ffici&#235;nten van een kwadratisch polynoom op, wat resulteert in het quoti&#235;nt 2x^2 &#8722; 2x &#8722; 2 en een rest van 0. Dit geeft aan dat x &#8722; 2 een factor is.
Deze uitkomst illustreert direct de Reststelling: de rest bij deling van een polynoom f(x) door x &#8722; c is gelijk aan f(c). Is die rest 0, dan bevestigt de Factorstelling dat x &#8722; c een factor van f(x) is.

Synthetische deling van polynomen

Synthetische deling is een effici&#235;nte, algoritmische methode om een polynoom te delen door een lineair binomium van de vorm x &#8722; c, waarbij 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.

Using Synthetic Division to Find Factors | JoVE K12

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Using Synthetic Division to Find Factors

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