(a + b)^n

Definition

The expression $(a + b)^n$ represents the expansion of a binomial raised to a positive integer power, where 'a' and 'b' are any numbers and 'n' is a non-negative integer. This concept is central to understanding how binomials can be expanded into a series of terms, each involving coefficients that are determined by the Binomial Theorem, allowing for efficient computation of powers of sums.

5 Must Know Facts For Your Next Test

1. $(a + b)^n$ can be expanded using the Binomial Theorem, which states that $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.
2. The binomial coefficients $inom{n}{k}$ can be calculated using the formula $rac{n!}{k!(n-k)!}$, where '!' denotes factorial.
3. The number of terms in the expansion of $(a + b)^n$ is always equal to $n + 1$.
4. When expanding $(a + b)^n$, the powers of 'a' decrease while the powers of 'b' increase across the terms.
5. The coefficients obtained from expanding $(a + b)^n$ correspond to the rows of Pascal's Triangle, making it easy to find them without computation.

Review Questions

• How does the Binomial Theorem apply to expand the expression $(a + b)^n$, and what role do binomial coefficients play in this expansion?
  • The Binomial Theorem provides a systematic way to expand $(a + b)^n$ into a sum of terms involving powers of 'a' and 'b'. Each term in this expansion is multiplied by a binomial coefficient, $inom{n}{k}$, which represents how many ways you can choose 'k' instances of 'b' from 'n' total factors. This makes it possible to calculate the expansion efficiently without needing to multiply out each term individually.
• Illustrate how Pascal's Triangle can be used to derive the coefficients for the expansion of $(a + b)^n$, including an example with a specific value of 'n'.
  • Pascal's Triangle visually represents the coefficients for any expansion of $(a + b)^n$. For example, if we take $n = 4$, we look at the 5th row (starting from row 0), which contains the coefficients: 1, 4, 6, 4, 1. Therefore, $(a + b)^4$ expands to $1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4$. Each coefficient corresponds to its position in Pascal's Triangle.
• Evaluate the significance of understanding $(a + b)^n$ in mathematical contexts beyond simple algebraic manipulation, particularly in combinatorics.
  • $(a + b)^n$ serves as a foundational concept in both algebra and combinatorics. Understanding this expansion not only aids in simplifying algebraic expressions but also connects deeply with combinatorial principles. For instance, binomial coefficients indicate how many ways there are to select elements from sets and have applications in probability and statistics. By mastering this concept, one gains tools for solving complex problems in various fields such as computer science and economics.

Related terms

Binomial Coefficient: The coefficient that appears in the expansion of $(a + b)^n$, often represented as $inom{n}{k}$, indicating the number of ways to choose 'k' elements from a set of 'n' elements.

Pascal's Triangle: A triangular array of numbers that provides the coefficients for the expansion of binomials. Each number is the sum of the two numbers directly above it, corresponding to the binomial coefficients.

Polynomial: An algebraic expression made up of terms, which can include constants, variables, and exponents. The expansion of $(a + b)^n$ results in a polynomial with multiple terms.