📚 Factorization: Common Methods and Typical Examples | 因式分解：常见方法与典型例题
Factorization, also called factoring, means writing a polynomial as a product of simpler non-constant polynomials. It is one of the most fundamental skills in algebra, because it appears in equation solving, simplification of rational expressions, graphing, and higher-level calculus. The goal of this article is to review the main factorization methods and to illustrate them with typical examples.
因式分解，也称分解因式，是指把一个多项式写成若干个更简单的非零次多项式的乘积。它是代数中最基础的技能之一，常用于解方程、化简分式、作图和后续微积分学习。本文的目的就是系统地复习因式分解的主要方法，并配以典型例题加以说明。
1. What Is Factorization? | 什么是因式分解？
To factor a polynomial means to transform an expression such as x² + 5x + 6 into a product like (x + 2)(x + 3). The original expression is expanded, while the factorized form is equivalent but written as multiplication.
分解一个多项式，就是把像 x² + 5x + 6 这样的表达式改写为 (x + 2)(x + 3) 这样的乘积。原表达式是展开形式，因式分解后的形式与原式等价，但写成乘法形式。
Factorization is the reverse process of expansion. When we expand (a + b)(c + d), we multiply the terms; when we factor, we reverse the process and look for common structures.
因式分解是展开的逆运算。展开 (a + b)(c + d) 时我们把各项相乘；因式分解时则反过来，寻找原式中的结构并写成乘积。
2. Extraction of Common Factors | 提公因式法
The first thing to check in any factorization problem is whether all terms share a common factor. If they do, write this factor outside the bracket and divide each term by it.
做因式分解时，第一步永远是检查各项是否有公因式。如果有，就把这个公因式提到括号外，并将每一项除以它。
ab + ac = a(b + c)
For example, consider 6x³y + 9x²y². The greatest common factor of the coefficients 6 and 9 is 3. For the variables, the lowest power of x is x², and the lowest power of y is y. Therefore the GCF is 3x²y.
例如，考虑 6x³y + 9x²y²。系数 6 和 9 的最大公因数是 3；对于字母部分，x 的最低次幂是 x²，y 的最低次幂是 y。因此最大公因式是 3x²y。
6x³y + 9x²y² = 3x²y(2x + 3y)
Always leave the remaining polynomial simple and check that no common factor remains inside the bracket.
分解后要注意括号内是否还能继续提取公因式，并检查是否已经提取彻底。
3. Grouping | 分组分解法
Grouping is useful when a polynomial has four or more terms. We split the expression into groups, factor each group separately, and then look for a common binomial factor.
当多项式有四项或更多项时，分组法非常有效。我们把表达式分成若干组，先分别对每组进行因式分解，再寻找共同的二项式因式。
ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b)
Take the expression x³ + 3x² − 4x − 12. Group the first pair and the second pair.
以 x³ + 3x² − 4x − 12 为例。先把前两项和后两项分别分组。
x³ + 3x² − 4x − 12 = x²(x + 3) − 4(x + 3) = (x + 3)(x² − 4)
We can factor further because x² − 4 is a difference of two squares.
我们还可以继续分解，因为 x² − 4 是平方差。
(x + 3)(x² − 4) = (x + 3)(x − 2)(x + 2)
If the first attempt at grouping does not produce a common binomial, try regrouping the terms in a different order.
如果第一次分组没有出现公共二项式，可以尝试重新调整项的顺序再分组。
4. Difference of Two Squares | 平方差公式
The difference of two squares is one of the most commonly used identities in secondary mathematics. It is important that the expression is a subtraction between two perfect squares.
平方差公式是中学数学中最常用的公式之一。使用它时必须确保原式是两个完全平方数的差。
a² − b² = (a + b)(a − b)
For example, 9x² − 16 can be written as (3x)² − 4². Thus we factor it as follows.
例如，9x² − 16 可以写成 (3x)² − 4²。因此可以这样分解。
9x² − 16 = (3x)² − 4² = (3x + 4)(3x − 4)
This formula can also be applied repeatedly. For example, x⁴ − 81 is first a difference of squares, and one of its factors is again a difference of squares.
这个公式可以反复使用。例如，x⁴ − 81 首先是平方差，而其中一个因式仍然是平方差。
x⁴ − 81 = (x² + 9)(x² − 9) = (x² + 9)(x + 3)(x − 3)
Notice that x² + 9 is a sum of squares, which cannot be factored using real coefficients into linear factors.
注意 x² + 9 是平方和，在实数范围内不能再分解为一次因式的乘积。
5. Perfect Square Trinomials | 完全平方公式
A perfect square trinomial comes from squaring a binomial. It has the form a² + 2ab + b² or a² − 2ab + b².
完全平方式是由二项式平方展开得到的。它的形式是 a² + 2ab + b² 或 a² − 2ab + b²。
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
term is (3y)², and the middle term is 2 · x · 3y = 6xy. Therefore it is a perfect square.
判断 x² + 6xy + 9y² 是否为完全平方式。首项是 x²，末项是 (3y)²，中间项是 2 · x · 3y = 6xy。因此它确实是完全平方式。
x² + 6xy + 9y² = (x + 3y)²
² − 20x + 25. Since 2x · 5 · 2 = 20x, we get a perfect square with a negative middle term.
另一个例子是 4x² − 20x + 25。因为 2x · 5 · 2 = 20x，所以它是中间项为负的完全平方式。
4x² − 20x + 25 = (2x − 5)²
Always test whether the middle term is exactly twice the product of the two square roots before using this formula.
使用完全平方公式前，一定要验证中间项是否正好等于两项平方根的乘积的二倍。
6. Sum and Difference of Two Cubes | 立方和与立方差
When a polynomial is a sum or a difference of two perfect cubes, we use the following formulas.
当一个多项式是两个完全立方数的和或差时，我们使用下面两个公式。
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
or 8x³ + 27, write it as (2x)³ + 3³. Taking a = 2x and b = 3, we obtain the factorization.
对于 8x³ + 27，把它写成 (2x)³ + 3³。取 a = 2x，b = 3，就可以分解。
8x³ + 27 = (2x + 3)(4x² − 6x + 9)
Similarly, x³ − 125 = x³ − 5³. Therefore it factors as a difference of cubes.
类似地，x³ − 125 = x³ − 5³，所以按立方差公式分解。
x³ − 125 = (x − 5)(x² + 5x + 25)
Be careful with the signs in the second factor. The middle term takes the opposite sign from the original sum or difference.
注意第二个因式中的符号。中间项的符号与原式中立方和或立方差的符号相反。
7. Factorising Quadratic Trinomials | 二次三项式分解法
For a quadratic trinomial of the form x² + px + q, we look for two numbers whose product is q and whose sum is p.
对于形如 x² + px + q 的二次三项式，我们需要
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