a/b 有理式计算器（求值、化简、渐近线、定义域）Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain)

使用说明与常见问题How-to & FAQ

「有理式计算器（求值、化简、渐近线、定义域）」是什么？What is the Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain) tool?

两种输入：线性分式 (ax + b)/(cx + d)，或者用系数列表输入任意多项式分子和分母。求在 x 处的值，用多项式最大公因式约分化简，列出使分母为 0 的排除值、零点、竖直渐近线、水平或斜渐近线以及“洞”（可去间断点）。本地处理。Two inputs: a linear fraction (ax + b)/(cx + d), or any polynomial numerator and denominator entered as coefficient lists. Evaluates at x, simplifies by cancelling the polynomial GCD, and lists excluded values, zeros, vertical asymptotes, the horizontal or slant asymptote, and holes (removable discontinuities). Processed locally.

如何使用「有理式计算器（求值、化简、渐近线、定义域）」？How do I use the Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain) tool?

在「有理式计算器（求值、化简、渐近线、定义域）」中，填写「输入方式」、「a」、「b」、「c」、「d」等参数，结果会随输入实时计算并显示在下方。In the Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain): Fill in "Input", "a", "b", "c", "d"; results are recalculated instantly as you type.

怎么化简有理式？How is a rational expression simplified?

把分子、分母分解因式后约去公因式。例如 (x² − 4)/(x − 2) = (x + 2)(x − 2)/(x − 2) = x + 2，但 x = 2 仍被排除，图像在 (2, 4) 处有一个“洞”。Factor the numerator and denominator and cancel common factors. (x² − 4)/(x − 2) = (x + 2)(x − 2)/(x − 2) = x + 2, but x = 2 stays excluded, leaving a hole at (2, 4).

渐近线怎么判断？How are the asymptotes found?

化简后分母为 0 的点是竖直渐近线。比较次数：分子次数小于分母时水平渐近线为 y = 0；相等时为首项系数之比；分子高 1 次时有斜渐近线（多项式除法的商）。After simplifying, zeros of the denominator give vertical asymptotes. Compare degrees: lower numerator degree gives y = 0, equal degrees give the ratio of leading coefficients, and a numerator one degree higher gives a slant asymptote (the quotient of polynomial division).

系数怎么输入？How do I enter coefficients?

从最高次到常数项依次输入，缺的次数填 0。例如 x³ + 1 输入 1 0 0 1，x² − 4 输入 1 0 −4。List them from the highest power down to the constant, using 0 for missing powers. x³ + 1 is 1 0 0 1 and x² − 4 is 1 0 −4.

能举个计算实例吗？Can you show a worked example?

例如在「有理式计算器（求值、化简、渐近线、定义域）」中填 输入方式 = 线性分式 (ax+b)/(cx+d)，a = 1，b = 0，c = 1，d = 0，x = 2，结果区显示：值（x = 2）：1；原式：(x) / (x)；化简：1；约去公因式：x；排除值（定义域外）：x ≠ 0；零点：无；竖直渐近线：无；洞（可去间断点）：(0, 1)；水平 / 斜渐近线：无（化简后是多项式 y = 1）For example, entering Input = Linear fraction (ax+b)/(cx+d), a = 1, b = 0, c = 1, d = 0, x = 2 in the Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain) gives: Value (x = 2): 1; Expression: (x) / (x); Simplified: 1; Common factor cancelled: x; Excluded values (not in domain): x ≠ 0; Zeros: none; Vertical asymptotes: none; Holes (removable): (0, 1); Horizontal / slant asymptote: none (it simplifies to the polynomial…

哪些输入会被检查？Which inputs are validated?

「有理式计算器（求值、化简、渐近线、定义域）」会检查这些输入——「a」、「b」、「c」、「d」、「x」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain) checks its inputs — "a", "b", "c", "d", "x": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「有理式计算器（求值、化简、渐近线、定义域）」的设置：「输入方式」可选 线性分式 (ax+b)/(cx+d)、多项式系数（默认 线性分式 (ax+b)/(cx+d)）；「a」默认 1；「b」默认 0；「c」默认 1；「d」默认 0；「分子系数（从最高次起）」默认 1 0 -4；「分母系数（从最高次起）」默认 1 -2；「x」默认 2。每次生成结果都按当前设置处理。Settings in the Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain) — "Input": Linear fraction (ax+b)/(cx+d), Polynomial coefficients (default Linear fraction (ax+b)/(cx+d)); "a" default 1; "b" default 0; "c" default 1; "d" default 0; "Numerator coefficients (highest power first)" default 1 0 -4; "Denominator coefficients (highest power first)" default 1 -2; "x" default 2. The result always uses the current settings.

计算精度如何？How precise are the results?

「有理式计算器（求值、化简、渐近线、定义域）」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 Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「有理式计算器（求值、化简、渐近线、定义域）」免费吗？需要注册吗？Is the Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain) free? Do I need an account?

「有理式计算器（求值、化简、渐近线、定义域）」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 Rational Expression Calculator (Evaluate, Simplify, Asymptotes, Domain) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.