📚 Example 7.4.1 | 例题7.4.1
This worked example explores a typical AQA A-level Mathematics problem involving the factor theorem. By examining the polynomial f(x) = x³ + ax² + bx − 6, we will determine the unknown coefficients a and b using the given factors and then write the cubic in fully factorised form.
这个例题探讨了一个典型的AQA A-level数学问题，应用因式定理。通过研究多项式 f(x) = x³ + ax² + bx − 6，我们将利用给定的因式求出未知系数 a 和 b，然后将三次多项式完全因式分解。
1. Understanding the Problem | 理解题目
Before any calculation, we must identify what is given and what is required. We have a cubic polynomial f(x) with two unknown coefficients, a and b. The condition that (x − 2) is a factor tells us that f(2) = 0. The condition f(1) = 0 tells us that (x − 1) is another factor. Our goal is to find a and b, and then factorise the cubic.
在计算之前，我们必须明确已知条件与要求。已知一个三次多项式 f(x) 含有两个未知系数 a 和 b。条件 (x − 2) 是因式意味着 f(2) = 0。条件 f(1) = 0 告诉我们 (x − 1) 是另一个因式。我们的目标是求出 a 和 b，然后对三次多项式进行因式分解。
2. Recapping the Factor Theorem | 因式定理回顾
The factor theorem states that for a polynomial f(x), if f(k) = 0, then (x − k) is a factor of f(x). Conversely, if (x − k) is a factor, then f(k) = 0. This theorem is essential for solving this question.
因式定理指出：对于多项式 f(x)，若 f(k) = 0，则 (x − k) 是 f(x) 的因式；反之，若 (x − k) 是因式，则 f(k) = 0。该定理在本题中至关重要。
3. Substituting x = 2 | 代入 x = 2
Since (x − 2) is a factor, we set x = 2. Substituting into f(x) gives f(2) = 0, so we can form an equation involving a and b:
因为 (x − 2) 是因式，我们令 x = 2。代入 f(x) 得到 f(2) = 0，从而建立含 a 和 b 的方程：
2³ + 4a + 2b − 6 = 0
Simplifying the constant terms leads to:
化简常数项后得到：
2a + b = −1
4. Substituting x =
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