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.
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- 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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Sashen littafi
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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