⅟ 部分分式分解计算器Partial Fraction Decomposition Calculator
把有理函数（多项式之比）分解成部分分式之和：支持一次因式、重复因式和不可约二次因式，分子次数不低于分母时先分离出多项式部分；结果给出普通写法和 LaTeX，并在多个点代入核对。Split a rational function (a ratio of polynomials) into a sum of partial fractions — linear, repeated and irreducible quadratic factors are all handled, and when the numerator's degree is not lower than the denominator's the polynomial part is separated out first. Results come in plain notation and LaTeX, checked numerically at several points.
使用说明与常见问题How-to & FAQ
「部分分式分解计算器」是什么？What is the Partial Fraction Decomposition Calculator tool?
把有理函数（多项式之比）分解成部分分式之和：支持一次因式、重复因式和不可约二次因式，分子次数不低于分母时先分离出多项式部分；结果给出普通写法和 LaTeX，并在多个点代入核对。Split a rational function (a ratio of polynomials) into a sum of partial fractions — linear, repeated and irreducible quadratic factors are all handled, and when the numerator's degree is not lower than the denominator's the polynomial part is separated out first. Results come in plain notation and LaTeX, checked numerically at several points.
如何使用「部分分式分解计算器」？How do I use the Partial Fraction Decomposition Calculator tool?
在「部分分式分解计算器」中，按需设置「有理函数」、「变量」等选项，然后点击「分解」按钮即可。In the Partial Fraction Decomposition Calculator: Set "Rational function", "Variable", then click "Decompose".
能分解哪些分式？Which fractions can it decompose?
分子和分母都必须是多项式（系数可以是整数、小数或分数）。分母会在有理数范围内因式分解，支持一次因式（x − 1）、重复因式（(x − 1)²）和不可约二次因式（x² + x + 1）；像 x² − 2 这样只能在实数范围内分解（需要 √2）的分母不会被拆开。Numerator and denominator must be polynomials (integer, decimal or fractional coefficients). The denominator is factored over the rationals, covering linear factors (x − 1), repeated factors ((x − 1)²) and irreducible quadratics (x² + x + 1). A denominator like x² − 2 that only splits over the reals (it needs √2) is left as is.
分子次数高于分母怎么办？What if the numerator's degree is higher?
分子次数不低于分母（假分式）时，会先做多项式除法，把多项式部分（如 x + 1）单独列出，剩下的真分式再分解。For an improper fraction (numerator degree not lower than the denominator's), polynomial division runs first: the polynomial part (such as x + 1) is shown separately and the remaining proper fraction is decomposed.
怎么确认结果正确？How do I know the result is right?
每次计算后都会在多个点把原式和分解结果分别代入求值；数值一致才显示“数值相等”，否则会给出提醒。也可以把结果通分，应能得到原来的分式。After each calculation, the original and the decomposition are both evaluated at several points; you see "agree" only when the values match, otherwise a caution is shown. You can also recombine the terms over a common denominator to get the original fraction back.
支持哪些写法？What syntax is supported?
四则运算 + - * /、乘方 ^、括号，函数 sin cos tan asin acos atan sinh cosh tanh sqrt abs exp log（自然对数，ln 也可以）等，常数 pi 和 e。乘号可以省略（2x、3sin(x)），也可以用 ×、÷、π、√。+ - * /, powers with ^, parentheses, functions such as sin cos tan asin acos atan sinh cosh tanh sqrt abs exp log (natural log; ln also works), and the constants pi and e. Multiplication can be implicit (2x, 3sin(x)); ×, ÷, π and √ are accepted too.
用的是什么计算引擎？What engine does it use?
开源符号计算库 nerdamer（Apache-2.0 许可），在独立的后台线程（Web Worker）中运行，超过 10 秒会自动停止，不会卡住页面；所有计算都在本地完成。The open-source symbolic library nerdamer (Apache-2.0), running in a background Web Worker that stops after 10 seconds so the page never freezes; everything is computed locally.
计算精度如何？How precise are the results?
「部分分式分解计算器」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Partial Fraction Decomposition Calculator uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「部分分式分解计算器」免费吗？需要注册吗？Is the Partial Fraction Decomposition Calculator free? Do I need an account?
「部分分式分解计算器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Partial Fraction Decomposition Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.