The Binomial Theorem
Bayani Gaba-gaba
Expanding \((a+b)^2\) is easy, and \((a+b)^3\) is manageable, but what about \((a+b)^{10}\)? Multiplying out by hand would be a nightmare. The binomial theorem gives the whole expansion at once, with coefficients you can read straight off using the combination numbers \(^nC_r\) you just learned.
In this lesson you will expand \((a+b)^n\) for a positive integer \(n\), see how the binomial coefficients control each term, and learn to pick out a single specific term using the general term. The neat example \((1+x)^4=1+4x+6x^2+4x^3+x^4\) shows the pattern clearly.
Manufura
- Use the binomial theorem for the expansion of (a + b) to the power n for positive integer n.
- Use the general term of the expansion, including finding a particular term.
Takardar Darasi
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Nazarin Darasi
Barka da kammala darasi akan The Binomial Theorem. 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 expansion of (1 + x)^4? A. 1 + 4x + 4x^2 + 4x^3 + x^4 B. 1 + 4x + 6x^2 + 4x^3 + x^4 C. 1 + 2x + 3x^2 + 2x^3 + x^4 D. 1 + 4x + 6x^2 + 4x^3 + 4x^4 Answer: B
- The binomial coefficients in the row for n = 4 are: A. 1, 3, 3, 1 B. 1, 4, 4, 1 C. 1, 4, 6, 4, 1 D. 1, 5, 10, 10, 5, 1 Answer: C
- In the expansion of (a + b)^n, the powers of a and b in each term add up to: A. r B. n - r C. n D. 1 Answer: C
- What is the term in x^2 in the expansion of (2 + x)^5? A. 10x^2 B. 40x^2 C. 80x^2 D. 160x^2 Answer: C
- The binomial coefficient that multiplies each term is: A. nPr B. nCr C. n! D. r! Answer: B
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