The expression $(x+y)^n$ represents the binomial expansion, which is the result of raising the sum of two variables, $x$ and $y$, to a positive integer power, $n$. This expression is a fundamental concept in the Binomial Theorem, a powerful tool for expanding and simplifying polynomial expressions.
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The Binomial Theorem states that $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$, where $n$ is a positive integer.
The coefficients in the binomial expansion are given by the binomial coefficients, $\binom{n}{k}$, which represent the number of ways to choose $k$ items from a set of $n$ items.
The exponents of $x$ and $y$ in the binomial expansion decrease and increase, respectively, as the index $k$ increases from $0$ to $n$.
The first and last terms in the binomial expansion are $x^n$ and $y^n$, respectively.
The Binomial Theorem can be used to expand and simplify polynomial expressions, as well as to calculate the probabilities of events in probability theory.
Review Questions
Explain the relationship between the binomial expansion $(x+y)^n$ and the Binomial Theorem.
The binomial expansion $(x+y)^n$ is a direct application of the Binomial Theorem, which provides a systematic way to expand a binomial expression raised to a positive integer power. The Binomial Theorem states that $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$, where the coefficients of the expansion are given by the binomial coefficients $\binom{n}{k}$. This formula allows for the efficient calculation of the terms in the binomial expansion, making it a powerful tool for simplifying and manipulating polynomial expressions.
Describe the role of binomial coefficients in the binomial expansion $(x+y)^n$.
The binomial coefficients, denoted as $\binom{n}{k}$, are a crucial component of the binomial expansion $(x+y)^n$. These coefficients represent the number of ways to choose $k$ items from a set of $n$ items, without regard to order. In the context of the binomial expansion, the binomial coefficients determine the coefficients of the terms in the expansion. Specifically, the coefficient of the term $x^{n-k} y^k$ is given by $\binom{n}{k}$. Understanding the properties and calculation of binomial coefficients is essential for working with the Binomial Theorem and the binomial expansion.
Analyze the pattern of the exponents in the binomial expansion $(x+y)^n$ and explain its significance.
The exponents of $x$ and $y$ in the binomial expansion $(x+y)^n$ follow a specific pattern: the exponent of $x$ decreases from $n$ to $0$, while the exponent of $y$ increases from $0$ to $n$. This pattern is a direct consequence of the Binomial Theorem formula, which states that $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$. The decreasing exponent of $x$ and the increasing exponent of $y$ reflect the fact that the binomial expansion is a sum of terms, each representing a different way of choosing $k$ items from a set of $n$ items. Understanding this pattern of exponents is crucial for recognizing the structure of the binomial expansion and for simplifying and manipulating polynomial expressions using the Binomial Theorem.
The Binomial Theorem is a formula that allows for the expansion of a binomial expression raised to a positive integer power, providing a systematic way to calculate the coefficients and exponents of the resulting polynomial.
The binomial coefficient, denoted as $\binom{n}{k}$, represents the number of ways to choose $k$ items from a set of $n$ items, without regard to order. It is a crucial component of the Binomial Theorem formula.
Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The numbers in this triangle are the binomial coefficients, which are used in the Binomial Theorem.