📚 Factorising | 因式分解
Factorising is the process of rewriting an algebraic expression as a product of simpler factors. It is one of the most important algebraic skills in Edexcel A-Level Mathematics, underpinning work on quadratics, polynomials, rational functions, calculus and curve sketching.
因式分解是将一个代数表达式改写为较简单因式乘积的过程。它是 Edexcel A-Level 数学中最重要的代数技能之一，是二次函数、多项式、有理函数、微积分和曲线作图等内容的基础。
1. Factorising Fundamentals | 因式分解基础
Factorising means expressing an algebraic expression as a product. For example, 6x + 9 can be written as 3(2x + 3). Factorising is the reverse of expanding brackets, so you can check any factorisation by multiplying out the factors.
因式分解意味着将一个代数式表示为乘积形式。例如，6x + 9 可以写成 3(2x + 3)。因式分解是展开括号的逆运算，因此你可以通过将因式乘开来检验任何因式分解。
In A-Level questions, you are often expected to factorise fully, which means continuing until no common factor or recognised pattern remains. Always look for the highest common factor first, then check for special forms such as quadratics or differences of squares.
在 A-Level 题目中，通常要求完全因式分解，即一直分解到没有公因式或可识别的模式为止。总是先寻找最高公因式，然后检查是否有特殊形式，如二次三项式或平方差。
2. Taking Out Common Factors | 提取公因式
The first step in any factorisation is to remove the highest common factor (HCF) of all terms. For example, 12x³ − 8x² + 4x has HCF 4x, so 12x³ − 8x² + 4x = 4x(3x² − 2x + 1).
任何因式分解的第一步都是提取所有项的最高公因式（HCF）。例如，12x³ − 8x² + 4x 的最高公因式为 4x，因此 12x³ − 8x² + 4x = 4x(3x² − 2x + 1)。
If the leading term is negative, it is often useful to take out a negative common factor. For instance, −5x² + 10x − 15 = −5(x² − 2x + 3). This can make the remaining quadratic easier to factorise.
如果首项为负，通常可以提取负公因式。例如，−5x² + 10x − 15 = −5(x² − 2x + 3)。这样可以使剩余的二次式更容易分解。
Always check whether a common factor remains inside the bracket after the first step. Missing a second common factor is one of the most common errors in factorisation.
始终检查第一步之后括号内是否仍有公因式。遗漏第二个公因式是因式分解中最常见的错误之一。
3. Factorising Quadratics: x² + bx + c | 二次三项式分解：x² + bx + c
A monic quadratic has the form x² + bx + c. To factorise it, find two numbers whose product is c and whose sum is b. If the numbers are p and q, then x² + bx + c = (x + p)(x + q).
首一二次三项式的形式为 x² + bx + c。要分解它，需找到两个数，它们的乘积为 c，和为 b。如果这两个数为 p 和 q，那么 x² + bx + c = (x + p)(x + q)。
For example, x² + 7x + 12 = (x + 3)(x + 4) because 3 × 4 = 12 and 3 + 4 = 7. Be careful with signs: x² − 5x + 6 = (x − 2)(x − 3), and x² − x − 12 = (x − 4)(x + 3).
例如，x² + 7x + 12 = (x + 3)(x + 4)，因为 3 × 4 = 12 且 3 + 4 = 7。注意符号：x² − 5x + 6 = (x − 2)(x − 3)，而 x² − x − 12 = (x − 4)(x + 3)。
| Quadratic | Factorised form |
|---|---|
| x² + 7x + 12 | (x + 3)(x + 4) |
| x² − 5x + 6 | (x − 2)(x − 3) |
| x² − x − 12 | (x − 4)(x + 3) |
The table above shows how the signs of b and c determine the signs inside the brackets. This is a quick way to check your factorisation.
上表展示了 b 和 c 的符号如何决定括号内的符号。这是快速检验因式分解的一种方法。
4. Factorising Quadratics: ax² + bx
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