±p/q 有理根计算器（候选根 + 逐一检验 + 分解）Rational Zeros Calculator (Candidates, Testing and Factoring)
按有理根定理列出所有可能的有理根 ±p/q（p 是常数项的因数，q 是首项系数的因数）。如果输入完整的整数系数多项式，还会逐一代入检验，用综合除法找出真正的有理根（含重根），并给出剩余因式。本地处理。Lists every possible rational zero ±p/q from the Rational Root Theorem (p divides the constant term, q divides the leading coefficient). Enter the full integer-coefficient polynomial to test each candidate, find the actual rational zeros with synthetic division (including repeated ones), and see the remaining factor. Processed locally.
使用说明与常见问题How-to & FAQ
「有理根计算器（候选根 + 逐一检验 + 分解）」是什么？What is the Rational Zeros Calculator (Candidates, Testing and Factoring) tool?
按有理根定理列出所有可能的有理根 ±p/q（p 是常数项的因数，q 是首项系数的因数）。如果输入完整的整数系数多项式，还会逐一代入检验，用综合除法找出真正的有理根（含重根），并给出剩余因式。本地处理。Lists every possible rational zero ±p/q from the Rational Root Theorem (p divides the constant term, q divides the leading coefficient). Enter the full integer-coefficient polynomial to test each candidate, find the actual rational zeros with synthetic division (including repeated ones), and see the remaining factor. Processed locally.
如何使用「有理根计算器（候选根 + 逐一检验 + 分解）」？How do I use the Rational Zeros Calculator (Candidates, Testing and Factoring) tool?
在「有理根计算器（候选根 + 逐一检验 + 分解）」中，填写「首项系数」、「常数项」、「完整系数（可选，从最高次起）」等参数，结果会随输入实时计算并显示在下方。In the Rational Zeros Calculator (Candidates, Testing and Factoring): Fill in "Leading coefficient", "Constant term", "Full coefficients (optional, highest power first)"; results are recalculated instantly as you type.
有理根定理是什么？What is the Rational Root Theorem?
整数系数多项式的任何有理根 p/q（最简形式）中，p 一定整除常数项，q 一定整除首项系数。所以只需检验有限个候选。For a polynomial with integer coefficients, any rational root p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient, so only finitely many candidates need checking.
候选一定是根吗？Are all candidates roots?
不是。候选只是可能的根，要代入或用综合除法检验。例如 x² − 2 的候选 ±1、±2 都不是根，它的根 ±√2 是无理数。No. They are only possible roots and must be tested by substitution or synthetic division. For x² − 2 none of ±1 and ±2 works; its roots ±√2 are irrational.
常数项为 0 怎么办？What if the constant term is 0?
先提出 x：例如 x³ − 2x² + x = x(x² − 2x + 1)，0 是一个根，再对剩下的多项式用定理。输入完整系数时本页自动处理。Factor out x first: x³ − 2x² + x = x(x² − 2x + 1), so 0 is a root, then apply the theorem to what remains. This page does it automatically when you enter the full coefficients.
常数项可以是负数吗？Can the constant term be negative?
可以。有理根定理只看常数项的绝对值 |a₀| 的因数：例如 x² − x − 6 的常数项为 −6，候选 p 为 ±1、±2、±3、±6，根是 −2 和 3。首项系数也可以为负。Yes. The Rational Root Theorem only uses the factors of |a₀|: for x² − x − 6 the constant term is −6, the candidates for p are ±1, ±2, ±3, ±6, and the zeros are −2 and 3. The leading coefficient may be negative too.
能举个计算实例吗？Can you show a worked example?
例如在「有理根计算器（候选根 + 逐一检验 + 分解）」中填 首项系数 = 1，常数项 = 6，结果区显示：候选根：-6, -3, -2, -1, 1, 2, 3, 6；候选数：8；p（常数项因数）：±1, ±2, ±3, ±6；q（首项系数因数）：±1For example, entering Leading coefficient = 1, Constant term = 6 in the Rational Zeros Calculator (Candidates, Testing and Factoring) gives: Candidates: -6, -3, -2, -1, 1, 2, 3, 6; Number of candidates: 8; p (factors of the constant): ±1, ±2, ±3, ±6; q (factors of the leading coefficient): ±1
哪些输入会被检查？Which inputs are validated?
「有理根计算器（候选根 + 逐一检验 + 分解）」会检查这些输入——「首项系数」：可留空；「常数项」：须为整数（可为负数；为 0 时 x 是一个因式）；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 Zeros Calculator (Candidates, Testing and Factoring) checks its inputs — "Leading coefficient": optional; "Constant term": must be a whole number (negative allowed; 0 means x is a factor); 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?
「有理根计算器（候选根 + 逐一检验 + 分解）」的设置：「首项系数」默认 2；「常数项」默认 6；「完整系数（可选，从最高次起）」可留空。每次生成结果都按当前设置处理。Settings in the Rational Zeros Calculator (Candidates, Testing and Factoring) — "Leading coefficient" default 2; "Constant term" default 6; "Full coefficients (optional, highest power first)" optional. The result always uses the current settings.
计算精度如何？How precise are the results?
「有理根计算器（候选根 + 逐一检验 + 分解）」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 Zeros Calculator (Candidates, Testing and Factoring) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「有理根计算器（候选根 + 逐一检验 + 分解）」免费吗？需要注册吗？Is the Rational Zeros Calculator (Candidates, Testing and Factoring) free? Do I need an account?
「有理根计算器（候选根 + 逐一检验 + 分解）」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 Zeros Calculator (Candidates, Testing and Factoring) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.