Binomial Expansion
Bayani Gaba-gaba
Raising a binomial to a high power by multiplying bracket after bracket is painfully slow. The binomial theorem replaces all that repeated multiplication with a single compact formula, letting you expand expressions like \((1 + x)^{10}\) or \((2 - 3x)^7\) term by term in seconds. It is one of the most versatile tools in algebra.
You will learn to use Pascal's triangle and the \(\binom{n}{r}\) notation for positive integer powers, then extend the theorem to rational exponents where the expansion becomes an infinite series. Along the way you will see how to extract individual coefficients, approximate awkward roots and powers, and judge when the series is valid. Every technique here pays dividends across the rest of further pure mathematics.
Manufura
- Use the binomial series (1 + x)^n when n is a positive integer
- Use the binomial series (1 + x)^n when n is rational and |x| < 1
- Understand the validity condition for the expansion when n is rational
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Barka da kammala darasi akan Binomial Expansion. Yanzu da kuka bincika mahimman ra'ayoyi da ra'ayoyi, 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.
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- What is the value of the binomial coefficient C(8,3)? A) 24 B) 56 C) 336 D) 6720 Answer: B
- How many terms are there in the expansion of (a + b)^9? A) 8 B) 9 C) 10 D) 18 Answer: C
- The expansion of (1 + x)^(-2) is valid when: A) x > 0 B) |x| < 1 C) x < 2 D) |x| < 2 Answer: B
- What is the coefficient of x^2 in the expansion of (1 + 3x)^5? A) 30 B) 45 C) 90 D) 270 Answer: C
- In the expansion of (1 + x)^n where n = 1/3, what is the second term? A) x/3 B) x C) 3x D) -x/3 Answer: A
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Ƙarin koyo daga tushe na buɗe
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Sashen littafi
Binomial Theorem
Apply the Binomial Theorem.
"College Algebra: 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/college-algebra/pages/9-6-binomial-theorem. Access for free at https://openstax.org/books/college-algebra/pages/1-introduction
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