fractional decomposition

简明释义

分级分解

英英释义

例句

1.In engineering, fractional decomposition helps in analyzing transfer functions.

在工程学中，分式分解有助于分析传递函数。

2.To solve the integral, we applied fractional decomposition to break it into simpler parts.

为了求解这个积分，我们应用了分式分解将其分解成更简单的部分。

3.The technique of fractional decomposition is essential in advanced algebra courses.

在高级代数课程中，分式分解技术是必不可少的。

4.In calculus, we often use fractional decomposition to simplify complex rational expressions.

在微积分中，我们常常使用分式分解来简化复杂的有理表达式。

5.Using fractional decomposition, we can express a function as a sum of simpler fractions.

通过使用分式分解，我们可以将一个函数表示为几个简单分数的和。

作文

In mathematics, particularly in the field of algebra, one often encounters the concept of fractional decomposition. This technique is essential for simplifying complex rational expressions into a sum of simpler fractions. Understanding fractional decomposition can greatly aid in solving integrals and differential equations, making it a valuable tool for students and professionals alike.The process of fractional decomposition involves breaking down a given rational function into a series of simpler fractions that are easier to work with. For instance, consider the rational expression 1/(x^2 - 1). To apply fractional decomposition, we first factor the denominator into (x - 1)(x + 1). This allows us to express the original fraction as a sum of two simpler fractions: A/(x - 1) + B/(x + 1), where A and B are constants that we need to determine.To find the values of A and B, we multiply both sides of the equation by the denominator (x^2 - 1) to eliminate the fractions. This results in the equation 1 = A(x + 1) + B(x - 1). Expanding this equation gives us a linear combination of x, which we can then solve by substituting convenient values for x. For example, if we let x = 1, we can easily solve for A, and if we let x = -1, we can solve for B.Once we have determined the values of A and B, we can rewrite the original expression using fractional decomposition. This simplification makes further calculations, such as integration or differentiation, much more manageable. For instance, integrating the simplified fractions A/(x - 1) and B/(x + 1) is straightforward compared to integrating the original rational function.Moreover, fractional decomposition is not limited to simple linear factors; it can also be applied to irreducible quadratic factors. For example, if we have a denominator of the form (x^2 + 1), the corresponding fractional decomposition would involve a term of the form (Ax + B)/(x^2 + 1). This versatility makes fractional decomposition a powerful technique in various areas of mathematics, including calculus, where it is frequently used to simplify integrals involving rational functions.In conclusion, the concept of fractional decomposition plays a crucial role in the simplification of rational expressions. By breaking down complex fractions into simpler components, mathematicians and students can tackle problems with greater ease. The ability to perform fractional decomposition effectively opens up new avenues for solving mathematical challenges, highlighting its importance in both academic and practical applications. As such, mastering this technique is essential for anyone looking to deepen their understanding of algebra and calculus.

在数学中，尤其是在代数领域，人们经常会遇到“分式分解”这一概念。这种技术对于将复杂的有理表达式简化为更简单的分数至关重要。理解“分式分解”可以大大帮助解决积分和微分方程，使其成为学生和专业人士都非常宝贵的工具。“分式分解”的过程涉及将给定的有理函数分解成一系列更简单的分数，从而使其更易于处理。例如，考虑有理表达式1/(x^2 - 1)。为了应用“分式分解”，我们首先将分母因式分解为(x - 1)(x + 1)。这使我们能够将原始分数表示为两个更简单分数的和：A/(x - 1) + B/(x + 1)，其中A和B是我们需要确定的常数。为了找到A和B的值，我们将方程两边乘以分母(x^2 - 1)以消除分数。这导致方程1 = A(x + 1) + B(x - 1)。扩展这个方程后，我们得到一个x的线性组合，我们可以通过替代方便的x值来求解。例如，如果我们让x = 1，我们可以很容易地求出A的值；如果我们让x = -1，我们可以求出B的值。一旦我们确定了A和B的值，就可以使用“分式分解”重新写出原始表达式。这种简化使得进一步的计算，例如积分或微分，变得更加可管理。例如，积分简化后的分数A/(x - 1)和B/(x + 1)要比积分原始有理函数简单得多。此外，“分式分解”不仅限于简单的线性因子；它还可以应用于不可约的二次因子。例如，如果我们有形式为(x^2 + 1)的分母，则相应的“分式分解”将涉及形式为(Ax + B)/(x^2 + 1)的项。这种多功能性使得“分式分解”在包括微积分在内的多个数学领域中成为一种强大的技术，因为它常常用于简化涉及有理函数的积分。总之，“分式分解”的概念在有理表达式的简化中发挥着至关重要的作用。通过将复杂的分数分解为更简单的组成部分，数学家和学生可以更轻松地解决问题。有效地执行“分式分解”的能力为解决数学挑战打开了新的途径，突显了其在学术和实际应用中的重要性。因此，掌握这一技术对任何希望加深对代数和微积分理解的人来说都是必不可少的。