The Binomial Theorem
Muhtasari
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.
Malengo
- 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.
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Hongera kwa kukamilisha somo la The Binomial Theorem. Sasa kwa kuwa umechunguza dhana na mawazo muhimu, ni wakati wa kuweka ujuzi wako kwa mtihani. Sehemu hii inatoa mazoezi mbalimbali maswali yaliyoundwa ili kuimarisha uelewaji wako na kukusaidia kupima ufahamu wako wa nyenzo.
Utakutana na mchanganyiko wa aina mbalimbali za maswali, ikiwemo maswali ya kuchagua jibu sahihi, maswali ya majibu mafupi, na maswali ya insha. Kila swali limebuniwa kwa umakini ili kupima vipengele tofauti vya maarifa yako na ujuzi wa kufikiri kwa makini.
Tumia sehemu hii ya tathmini kama fursa ya kuimarisha uelewa wako wa mada na kubaini maeneo yoyote ambapo unaweza kuhitaji kusoma zaidi. Usikatishwe tamaa na changamoto zozote utakazokutana nazo; badala yake, zitazame kama fursa za kukua na kuboresha.
- 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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Sehemu ya kitabu
Binomial Theorem
Apply the Binomial Theorem.
"Algebra and Trigonometry: Binomial Theorem" 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/13-6-binomial-theorem. Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction
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