Binomial Expansion
Gbogbo ọrọ náà
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.
Ebumnobi
- 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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Ekele diri gi maka imecha ihe karịrị na Binomial Expansion. Ugbu a na ị na-enyochakwa isi echiche na echiche ndị dị mkpa, ọ bụ oge iji nwalee ihe ị ma. Ngwa a na-enye ụdị ajụjụ ọmụmụ dị iche iche emebere iji kwado nghọta gị wee nyere gị aka ịmata otú ị ghọtara ihe ndị a kụziri.
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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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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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