× 多项式乘法计算器（任意多项式展开合并）Multiplying Polynomials Calculator (Expand and Combine Any Polynomials)

使用说明与常见问题How-to & FAQ

「多项式乘法计算器（任意多项式展开合并）」是什么？What is the Multiplying Polynomials Calculator (Expand and Combine Any Polynomials) tool?

输入两个多项式（如 2x + 3、x^2 − 4x + 1，可用任意一个字母作变量），按分配律逐项相乘并合并同类项，给出展开结果、次数、系数列表、逐项相乘的过程，以及代入 x 值后的结果。本地处理。Enter two polynomials (such as 2x + 3 and x^2 − 4x + 1, using any single letter as the variable) to multiply every term by every term and combine like terms, with the expanded result, its degree, the coefficient list, the term-by-term products and the value at a chosen x. Processed locally.

如何使用「多项式乘法计算器（任意多项式展开合并）」？How do I use the Multiplying Polynomials Calculator (Expand and Combine Any Polynomials) tool?

在「多项式乘法计算器（任意多项式展开合并）」中，填写「第一个多项式」、「第二个多项式」、「代入 x 的值（可选）」等参数，结果会随输入实时计算并显示在下方。In the Multiplying Polynomials Calculator (Expand and Combine Any Polynomials): Fill in "First polynomial", "Second polynomial", "Value of x (optional)"; results are recalculated instantly as you type.

多项式怎么相乘？How are polynomials multiplied?

用分配律把第一个多项式的每一项分别乘以第二个多项式的每一项，指数相加、系数相乘，最后合并同类项。Use the distributive law: multiply every term of the first by every term of the second, adding exponents and multiplying coefficients, then combine like terms.

怎么输入？How do I type them?

用 ^ 表示乘方，例如 3x^2 - 2x + 1；系数和变量之间不用写乘号。也可以用 y、t 等其他字母，但两个多项式要用同一个字母。Use ^ for powers, as in 3x^2 - 2x + 1, with no multiplication sign between coefficient and variable. Other letters such as y or t work, but both polynomials must use the same one.

乘积的次数是多少？What is the degree of the product?

两个非零多项式乘积的次数等于它们次数之和，例如二次式乘一次式得到三次式。The degree of the product of two non-zero polynomials is the sum of their degrees, so a quadratic times a linear gives a cubic.

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

例如在「多项式乘法计算器（任意多项式展开合并）」中填 第一个多项式 = 2x + 3，第二个多项式 = 4x + 5，代入 x 的值（可选） = 1，结果区显示：乘积：(2x + 3)(4x + 5) = 8x² + 22x + 15；次数：2；系数（从高到低）：8, 22, 15；系数和（x = 1 时的值）：45；逐项相乘：(3)(5) = 15(3)(4x) = 12x(2x)(5) = 10x(2x)(4x) = 8x²；x = 1 时：45For example, entering First polynomial = 2x + 3, Second polynomial = 4x + 5, Value of x (optional) = 1 in the Multiplying Polynomials Calculator (Expand and Combine Any Polynomials) gives: Product: (2x + 3)(4x + 5) = 8x² + 22x + 15; Degree: 2; Coefficients (high to low): 8, 22, 15; Sum of coefficients (value at x = 1): 45; Term-by-term products: (3)(5) = 15(3)(4x) = 12x(2x)(5) = 10x(2x)(4x) = 8x²; At x = 1: 45

哪些输入会被检查？Which inputs are validated?

「多项式乘法计算器（任意多项式展开合并）」会检查这些输入——「代入 x 的值（可选）」：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 Multiplying Polynomials Calculator (Expand and Combine Any Polynomials) checks its inputs — "Value of x (optional)": 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?

「多项式乘法计算器（任意多项式展开合并）」的设置：「第一个多项式」默认 2x + 3；「第二个多项式」默认 4x + 5；「代入 x 的值（可选）」默认 1。每次生成结果都按当前设置处理。Settings in the Multiplying Polynomials Calculator (Expand and Combine Any Polynomials) — "First polynomial" default 2x + 3; "Second polynomial" default 4x + 5; "Value of x (optional)" default 1. The result always uses the current settings.

计算精度如何？How precise are the results?

「多项式乘法计算器（任意多项式展开合并）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字），部分功能使用 BigInt 处理大整数。The Multiplying Polynomials Calculator (Expand and Combine Any Polynomials) uses double-precision floating point (about 15–16 significant digits); some features use BigInt for large integers.

「多项式乘法计算器（任意多项式展开合并）」免费吗？需要注册吗？Is the Multiplying Polynomials Calculator (Expand and Combine Any Polynomials) free? Do I need an account?

「多项式乘法计算器（任意多项式展开合并）」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 Multiplying Polynomials Calculator (Expand and Combine Any Polynomials) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.