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

Sentetik bölme, polinomları yalnızca katsayılarını kullanarak bölmek için basitleştirilmiş bir yöntemdir. Bölen, c'nin gerçek bir sayı olduğu x - c biçiminde doğrusal bir binom olduğunda kullanılır.Süreç, temettü katsayılarının azalan derece sırasına göre yazılması ve baştaki terim ile sabit arasındaki eksik dereceler için sıfırların eklenmesiyle başlar. Bölenden gelen c sabiti, bölme düzeninin sol tarafına yerleştirilir.İlk katsayı satırın altına yazılır. Daha sonra sabit ile çarpılır ve sonuç bir sonraki katsayıya eklenir. Bu çarpma ve toplama döngüsü satır boyunca devam eder.Satırdaki son değer kalanı temsil ederken, önceki değerler bölümün katsayılarını oluşturur - temettünün derecesinden bir küçük olan bir polinom. Nihai sonuç, temettü eşittir bölen çarpı bölüm artı kalan olarak ifade edilebilir.İş dünyasında polinomlar, üretim düzeyine göre karı modelleyebilir. Belirli bir seviyedeki karı bulmak için, polinom bir binom x eksi o seviyeye bölünür.Bu bölme için sentetik bölme yöntemini kullanmak, o seviyedeki karı temsil eden kalanı verir.

polinomların sentetik bölmesi

Synthetic Disvision of Polynomials

Sentetik b&#246;lme, bir polinomu x - c bi&#231;imindeki do&#287;rusal bir binomla b&#246;lmek i&#231;in etkili bir algoritmik yakla&#351;&#305;md&#305;r; burada c bir ger&#231;ek say&#305;d&#305;r. Bu y&#246;ntem, polinomlar&#305;n geleneksel uzun b&#246;lmede yer alan zahmetli ad&#305;mlardan ka&#231;&#305;nan basitle&#351;tirilmi&#351; s&#252;reci sayesinde faydal&#305;d&#305;r. Hesaplamay&#305; basitle&#351;tirir ve polinomlar&#305; de&#287;erlendirmek ve &#231;arpanlar&#305;n&#305; belirlemek i&#231;in pratik bir ara&#231; g&#246;revi g&#246;r&#252;r.
Sentetik b&#246;lme i&#351;lemi i&#231;in, b&#246;l&#252;nen polinomun katsay&#305;lar&#305;n&#305; azalan kuvvetlere g&#246;re listeleyerek ba&#351;lan&#305;r. Herhangi bir terim eksikse, hizalamay&#305; korumak i&#231;in katsay&#305;lar&#305; s&#305;f&#305;r olarak kaydedilir. x - c b&#246;leninden gelen c sabiti, sentetik b&#246;lme &#351;emas&#305;n&#305;n soluna yerle&#351;tirilir. &#304;&#351;lem, de&#287;i&#351;tirilmeden a&#351;a&#287;&#305;ya indirilen ba&#351; katsay&#305; ile ba&#351;lar. Bu de&#287;er c ile &#231;arp&#305;l&#305;r ve sonu&#231; bir sonraki katsay&#305;ya eklenir. Bu &#231;arpma ve toplama dizisi, t&#252;m katsay&#305;lar i&#351;lenene kadar devam eder.
Son sat&#305;rda &#252;retilen say&#305;lar, sondaki kalan de&#287;er hari&#231;, b&#246;l&#252;m polinomunun katsay&#305;lar&#305;n&#305; temsil eder. &#214;rne&#287;in, 2x^3 - 6x^2 + 2x - 4'&#252; sentetik b&#246;lme i&#351;lemi kullanarak x - 2'ye b&#246;lmek, ikinci dereceden bir polinomun katsay&#305;lar&#305;n&#305; verir ve 2x^2 - 2x - 2'nin b&#246;l&#252;m&#252;n&#252; ve s&#305;f&#305;r kalan&#305;n&#305; verir; bu da x - 2'nin bir &#231;arpan oldu&#287;unu g&#246;sterir.
Bu sonu&#231;, bir f(x) polinomunun x - c'ye b&#246;l&#252;m&#252;nden kalan&#305;n f(c) oldu&#287;unu belirten Kalan Teoremi'ni do&#287;rudan g&#246;stermektedir. Bu kalan s&#305;f&#305;rsa, &#199;arpan Teoremi, x - c'nin f(x)'in bir &#231;arpan&#305; oldu&#287;unu do&#287;rular.

Polinomların Sentetik Bölmesi

Sentetik b&#246;lme, bir polinomu x - c bi&#231;imindeki do&#287;rusal bir binomla b&#246;lmek i&#231;in etkili bir algoritmik yakla&#351;&#305;md&#305;r; ...

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Math Fundamentals

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Polynomial and Rational Functions

Polinom ve Rasyonel Fonksiyonlar

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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

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

Using Synthetic Division to Find Factors

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