The Factor Theorem | 因式定理

📚 The Factor Theorem | 因式定理

In Edexcel A Level Mathematics, the factor theorem is a fundamental result that connects the factors of a polynomial with the roots of the corresponding polynomial equation. It gives a quick algebraic test for deciding whether a linear expression such as (x – a) divides a polynomial f(x) exactly, and it is essential for factorising cubics and higher-degree polynomials.

在 Edexcel A Level 数学中，因式定理是一个把多项式的因式与对应多项式方程的根联系起来的基本结论。它提供了一种快速代数检验方法，用来判断一次式（如 (x – a)）是否整除多项式 f(x)，也是分解三次及更高次多项式的关键工具。

1. What is the Factor Theorem? | 什么是因式定理

For a polynomial f(x), the factor theorem states that (x – a) is a factor of f(x) if and only if f(a) = 0. Equivalently, if a is a root of the equation f(x) = 0, then (x – a) is a factor; conversely, if (x – a) is a factor, then f(a) = 0.

对于多项式 f(x)，因式定理指出：(x – a) 是 f(x) 的因式，当且仅当 f(a) = 0。等价地说，如果 a 是方程 f(x) = 0 的根，那么 (x – a) 就是因式；反过来，如果 (x – a) 是因式，那么 f(a) = 0。

Here f(a) means the value obtained by replacing x with a in the polynomial. For example, if f(x) = x³ – 4x² + x + 6, then f(2) = 8 – 16 + 2 + 6 = 0, so (x – 2) is a factor.

这里 f(a) 表示将 x 替换为 a 后得到的值。例如，若 f(x) = x³ – 4x² + x + 6，则 f(2) = 8 – 16 + 2 + 6 = 0，因此 (x – 2) 是它的因式。

(x – a) is a factor of f(x) ⇔ f(a) = 0

2. From Remainder Theorem to Factor Theorem | 从余数定理到因式定理

The factor theorem is a special case of the remainder theorem. The remainder theorem says that when a polynomial f(x) is divided by a linear divisor (x – a), the remainder is f(a).

因式定理是余数定理的特例。余数定理指出，当多项式 f(x) 除以一次除式 (x – a) 时，余数等于 f(a)。

Therefore, the division is exact (remainder 0) exactly when f(a) = 0. In that case, (x – a) must be a factor, which is precisely the factor theorem.

因此，当且仅当 f(a) = 0 时，除法是整除（余数为 0）。此时 (x – a) 必定是因式，这正是因式定理。

For instance, if f(x) = 2x³ – x² – 7x + 6, dividing by (x – 2) gives remainder f(2) = 16 – 4 – 14 + 6 = 4, so (x – 2) is not a factor. But f(1) = 2 – 1 – 7 + 6 = 0, so (x – 1) is a factor.

例如，若 f(x) = 2x³ – x² – 7x + 6，除以 (x – 2) 的余数为 f(2) = 16 – 4 – 14 + 6 = 4，所以 (x – 2) 不是因式；但 f(1) = 2 – 1 – 7 + 6 = 0，所以 (x – 1) 是因式。

3. Proving the Factor Theorem | 证明因式定理

The proof uses the division algorithm. For any polynomial f(x) and any constant a, we can write f(x) = (x – a)Q(x) + R, where Q(x) is the quotient polynomial and R is a constant remainder.

证明使用多项式除法算法。对任意多项式 f(x) 和常数 a，都可以写成 f(x) = (x – a)Q(x) + R，其中 Q(x) 是商多项式，R 是常数余数。

Substituting x = a gives f(a) = (a – a)Q(a) + R = 0 + R = R. Hence R = f(a).

代入 x = a 得 f(a) = (a – a)Q(a) + R = 0 + R = R，所以 R = f(a)。

If f(a) = 0, then R = 0 and f(x) = (x – a)Q(x), so (x – a) divides f(x) exactly. Conversely, if (x – a) is a factor, then R = 0 and f(a) = 0.

如果 f(a) = 0，则 R = 0，且 f(x) = (x – a)Q(x)，所以 (x – a) 整除 f(x)。反过来，如果 (x – a) 是因式，则 R = 0，因此 f(a) = 0。

4. Testing Linear Factors | 检验一次因式

For a polynomial with integer coefficients, the factor theorem can be combined with the rational root theorem. In Edexcel exams, you are usually given one factor or a value such as f(2) = 0, or you test small integer values like ±1, ±2, ±3.

对于整系数多项式，因式定理常与有理根定理结合使用。在 Edexcel 考试中，通常会给出一个因式或某个值（如 f(2) = 0），或者需要检验 ±1、±2、±3 等较小的整数值。

To test whether (x – a) is a factor, calculate f(a). If the result is zero, it is a factor; if not, it is not a factor. Do not write f(a) = 0 unless the calculation actually gives zero.

要检验 (x – a) 是否为因式，计算 f(a)。如果结果为零，则是因式；如果不是，则不是因式。除非计算结果确实为零，否则不要写 f(a) = 0。

Example: f(x) = x³ – 2x² – 5x + 6. Test a = 1: f(1) = 1 – 2 – 5 + 6 = 0, so (x – 1) is a factor. Test a = -1: f(-1) = -1 – 2 + 5 + 6 = 8, so (x + 1) is not a factor.

示例：f(x

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