Cubic Equations | 三次方程

📚 Cubic Equations | 三次方程

Cubic equations are polynomial equations of degree three and form an essential topic in AQA A-Level Mathematics. They appear in pure mathematics, coordinate geometry, and even modelling problems in mechanics and statistics. A thorough understanding of how to solve and interpret cubic equations lays a solid foundation for higher-level studies in calculus and numerical methods.

三次方程是次数为三的多项式方程，是 AQA A-Level 数学中一个核心课题。三次方程在纯数学、坐标几何乃至力学与统计中的建模问题里均有广泛出现。深入掌握三次方程的解法与图像性质，能为后续微积分和数值方法的学习打下坚实基础。

1. Definition and Standard Form | 定义与标准形式

A cubic equation can be written in the general form:

ax³ + bx² + cx + d = 0, where a, b, c, d are constants and a ≠ 0

三次方程的一般形式为：

ax³ + bx² + cx + d = 0，其中 a、b、c、d 为常数，且 a ≠ 0

The highest power of x is 3, which defines the equation’s degree. The coefficient a is known as the leading coefficient, and d is the constant term. If a = 0, the equation reduces to a quadratic, so the condition a ≠ 0 is essential.

变量 x 的最高次数为 3，这决定了方程的次数。系数 a 称为首项系数，d 称为常数项。若 a = 0，方程将退化为二次方程，因此 a ≠ 0 是三次方程的必备条件。

Cubic equations always have at least one real root. This follows from the Intermediate Value Theorem: as x → −∞ and x → +∞, the values of a cubic polynomial take opposite signs, so the curve must cross the x-axis at least once. Over the complex numbers, a cubic equation has exactly three roots, counted with multiplicity.

三次方程至少有一个实数根。这是介值定理的直接推论：当 x → −∞ 与 x → +∞ 时，三次多项式的值异号，因此曲线至少与 x 轴相交一次。在复数范围内，三次方程恰好有三个根（按重数计）。

2. The Factor Theorem | 因式定理

The factor theorem is one of the most powerful tools for solving cubic equations. It states that if f(p) = 0 for a polynomial f(x), then (x − p) is a factor of f(x). Conversely, if (x − p) is a factor, then f(p) = 0.

因式定理是求解三次方程

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）