Remainder And Factor Theorems
Bayani Gaba-gaba
Dividing one polynomial by another can be slow. The remainder and factor theorems give you a shortcut so powerful it feels like cheating: you can find the remainder of a division, or test whether something is a factor, just by substituting a single number. No long division required.
In this lesson you will meet the remainder theorem, which says the remainder when a polynomial is divided by \((x-a)\) is simply \(f(a)\), and the factor theorem, its special case when that remainder is zero. Together they are the key to factorising and solving cubic equations, a staple of the higher-level syllabus.
Manufura
- Know and use the remainder and factor theorems.
- Find factors of polynomials and solve cubic equations.
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Nazarin Darasi
Barka da kammala darasi akan Remainder And Factor Theorems. Yanzu da kuka bincika mahimman raayoyi da raayoyi, lokaci yayi da zaku gwada ilimin ku. Wannan sashe yana ba da ayyuka iri-iri Tambayoyin da aka tsara don ƙarfafa fahimtar ku da kuma taimaka muku auna fahimtar ku game da kayan.
Za ka gamu da haɗe-haɗen nau'ikan tambayoyi, ciki har da tambayoyin zaɓi da yawa, tambayoyin gajeren amsa, da tambayoyin rubutu. Kowace tambaya an ƙirƙira ta da kyau don auna fannoni daban-daban na iliminka da ƙwarewar tunani mai zurfi.
Yi wannan ɓangaren na kimantawa a matsayin wata dama don ƙarfafa fahimtarka kan batun kuma don gano duk wani yanki da kake buƙatar ƙarin karatu. Kada ka yanke ƙauna da duk wani ƙalubale da ka fuskanta; maimakon haka, ka kallesu a matsayin damar haɓaka da ingantawa.
- What is the remainder when f(x) = x^3 - 2x^2 + 5x - 1 is divided by (x - 2)? A. 0 B. 5 C. 9 D. 11 Answer: C
- By the factor theorem, (x - a) is a factor of f(x) when: A. f(0) = a B. f(a) = 0 C. f(a) = 1 D. f(1) = a Answer: B
- Is (x - 1) a factor of f(x) = x^3 - 6x^2 + 11x - 6? A. Yes, because f(1) = 0 B. No, because f(1) = 6 C. Yes, because f(0) = -6 D. No, because f(1) = 1 Answer: A
- The full factorisation of x^3 - 6x^2 + 11x - 6 is: A. (x - 1)(x - 2)(x - 3) B. (x + 1)(x + 2)(x + 3) C. (x - 1)^3 D. (x - 2)(x - 3)(x - 6) Answer: A
- What is the remainder when f(x) = x^2 + 3x + 2 is divided by (x + 2)? A. 0 B. 2 C. 6 D. 12 Answer: A
Yi aikin wadannan tambayoyi a cikin manhaja
Yi aikin wadannan tambayoyi a cikin manhaja
Littattafan da ake ba da shawarar karantawa
Ƙarin koyo daga tushe na buɗe
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Sashen littafi
Zeros of Polynomial Functions
Evaluate a polynomial using the Remainder Theorem.
"Algebra and Trigonometry: Zeros of Polynomial Functions" by OpenStax, OpenStax, Rice University, licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Source: https://openstax.org/books/algebra-and-trigonometry/pages/5-5-zeros-of-polynomial-functions. Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction
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