纯数 A — 代数与多项式

Pure A: Algebra & Polynomials

考点概述

因式定理、复杂因式分解、不等式证明

STEP 2/3 分开核验。复数与 2×2 矩阵可属 STEP 2；De Moivre、一般复根、平面与向量积等属 STEP 3。

学习重点

以下例题重点在于理解解题思路，建议先尝试自己解，再对照讲解。

1范围内书面例题

Let $f(x) = x^3 + px + q$ where $p$ and $q$ are real constants.

Show that $f(x)$ has a repeated root if and only if $4p^3 + 27q^2 = 0$ .

Show that $f(x) = x^3 - 3x - 2$ has a repeated root, and find all roots of $f$ .

Find all values of $k$ for which $x^3 - 3x - 2 = k$ has exactly two distinct real solutions.

Sketch $y = x^3 - 3x - 2$ and use your sketch to determine the number of real solutions of $x^3 - 3x - 2 = k$ for general $k$ .

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