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Binomial Theorem

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Algebra and Trigonometry

Definition

The Binomial Theorem provides a formula for expanding binomials raised to any positive integer power. It expresses $(a + b)^n$ as a sum involving terms of the form $C(n,k) \cdot a^{n-k} \cdot b^k$.

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5 Must Know Facts For Your Next Test

  1. The Binomial Theorem uses binomial coefficients, represented as $C(n,k)$ or $\binom{n}{k}$, which are calculated using factorials: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
  2. Each term in the expansion of $(a+b)^n$ is given by $C(n,k) \cdot a^{n-k} \cdot b^k$, where $k$ ranges from 0 to n.
  3. The sum of the exponents in each term is always equal to the power n: $(a+b)^5$ will have terms like $a^5$, $a^4b$, up to $b^5$.
  4. The binomial expansion is symmetric; for example, the coefficients in the expansion of $(a+b)^6$ are the same in reverse order.
  5. The Binomial Theorem can be used to compute probabilities in binomial distributions and solve problems involving combinations.

Review Questions

  • What is the general term for the expansion of $(x+y)^7$?
  • How do you calculate a binomial coefficient?
  • Explain the significance of symmetry in binomial expansions.
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