binomial theorem

简明释义

1. 二项式定理; 2. 二项展开式;

英英释义

例句

1.The expansion of (1 + x)^n using the binomial theorem 二项式定理 gives us a polynomial in x.

使用 binomial theorem 二项式定理 展开的 (1 + x)^n 给我们一个关于 x 的多项式。

2.The binomial theorem 二项式定理 is crucial in probability theory for calculating binomial distributions.

在概率论中，binomial theorem 二项式定理 对于计算二项分布至关重要。

3.In algebra, the binomial theorem 二项式定理 allows us to expand expressions like (a + b)^n.

在代数中，binomial theorem 二项式定理 使我们能够展开像 (a + b)^n 这样的表达式。

4.Using the binomial theorem 二项式定理, we can find the coefficients of each term in the expansion of (x + y)^5.

利用 binomial theorem 二项式定理，我们可以找到 (x + y)^5 展开中每一项的系数。

5.Many combinatorial problems can be solved using the binomial theorem 二项式定理 to determine the number of ways to choose items.

许多组合问题可以通过使用 binomial theorem 二项式定理 来确定选择物品的方式数量来解决。

作文

The binomial theorem is a fundamental principle in algebra that describes the expansion of powers of binomials. A binomial is simply an algebraic expression that contains two terms, such as (a + b). The binomial theorem provides a formula to expand expressions like (a + b)^n, where n is a non-negative integer. This theorem is not only essential for theoretical mathematics but also has practical applications in various fields such as physics, engineering, and economics.To understand the binomial theorem, let's consider its formula. The expansion of (a + b)^n can be expressed as:(a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n, and (n choose k) is the binomial coefficient, calculated as n! / (k!(n-k)!). This formula indicates that each term in the expansion consists of a coefficient multiplied by a power of 'a' and a power of 'b'. The coefficients correspond to the number of ways to choose k items from n items, which is why they are called binomial coefficients.One of the most remarkable aspects of the binomial theorem is its ability to simplify calculations involving large powers. For instance, expanding (x + y)^5 using the theorem gives us:(x + y)^5 = 1*x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + 1*y^5.This expansion shows all the possible combinations of x and y raised to different powers, along with their respective coefficients. Without the binomial theorem, calculating these values manually would be tedious and prone to error.The binomial theorem also has connections to combinatorics, which is the branch of mathematics dealing with combinations and permutations. The coefficients in the expansion directly relate to the number of ways to choose subsets of items, making this theorem a bridge between algebra and combinatorial mathematics. For example, in probability theory, the binomial theorem plays a crucial role in determining the likelihood of a certain number of successes in a fixed number of independent trials, each with the same probability of success.In addition to its mathematical significance, the binomial theorem has practical implications in real-world scenarios. Engineers use it to model systems and predict outcomes in various fields, including signal processing and structural analysis. Economists apply the theorem to forecast trends and analyze data, demonstrating its versatility beyond pure mathematics.In conclusion, the binomial theorem is an invaluable tool in mathematics that simplifies the process of expanding binomials and connects various mathematical concepts. Its applications span across multiple disciplines, showcasing its importance in both theoretical and practical contexts. Understanding the binomial theorem not only enhances one's mathematical skills but also opens doors to exploring more complex topics in algebra and beyond. As students and professionals alike delve into the world of mathematics, the binomial theorem remains a cornerstone of learning and application, highlighting the beauty and utility of mathematical principles in everyday life.

二项式定理是代数中的一个基本原理，描述了二项式的幂的展开。二项式只是一个包含两个项的代数表达式，例如(a + b)。二项式定理提供了一种公式，用于展开像(a + b)^n这样的表达式，其中n是一个非负整数。这个定理不仅对理论数学至关重要，而且在物理学、工程学和经济学等多个领域都有实际应用。要理解二项式定理，让我们考虑它的公式。 (a + b)^n的展开可以表示为：(a + b)^n = Σ (n choose k) * a^(n-k) * b^k，其中k从0到n变化，(n choose k)是二项式系数，计算方法为n! / (k!(n-k)!)。这个公式表明，展开中的每一项都由一个系数乘以'a'和'b'的幂组成。系数对应于从n个项目中选择k个项目的方法数，这就是它们被称为二项式系数的原因。二项式定理最引人注目的方面之一是它能够简化涉及大幂的计算。例如，使用此定理展开(x + y)^5给我们：(x + y)^5 = 1*x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + 1*y^5。这个展开显示了x和y的所有可能组合，并将它们提升到不同的幂次，以及相应的系数。如果没有二项式定理，手动计算这些值将是乏味且容易出错的。二项式定理还与组合数学有联系，组合数学是研究组合和排列的数学分支。展开中的系数直接与选择项目子集的方法数相关，使得这个定理成为代数和组合数学之间的桥梁。例如，在概率论中，二项式定理在确定固定次数的独立试验中成功的数量的可能性方面发挥着关键作用，每次试验都有相同的成功概率。除了其数学意义外，二项式定理在现实世界场景中也具有实际意义。工程师使用它来建模系统并预测各种领域的结果，包括信号处理和结构分析。经济学家应用该定理来预测趋势和分析数据，展示了它在纯数学之外的多样性。总之，二项式定理是数学中一个无价的工具，它简化了展开二项式的过程，并连接了各种数学概念。它的应用跨越多个学科，展示了它在理论和实践背景下的重要性。理解二项式定理不仅增强了一个人的数学技能，还为探索代数及更复杂主题打开了大门。当学生和专业人士深入数学世界时，二项式定理仍然是学习和应用的基石，突显了数学原理在日常生活中的美丽和实用性。