📚 Roots of Polynomial Equations | 多项式方程的根
For AQA A-Level Mathematics, the topic of roots of polynomial equations asks a central question: if you are given the coefficients of a polynomial, what can you say about its roots without solving it completely? This revision guide covers the symmetric relationships between roots and coefficients for quadratics, cubics and quartics, and shows how to use these relationships to simplify expressions, prove identities, and construct new polynomial equations.
在 AQA A-Level 数学中，多项式方程的根这一主题提出了一个核心问题：如果已知多项式的系数，你能在不完全求解方程的情况下判断出关于根的哪些信息？本复习指南涵盖二次、三次和四次多项式的根与系数之间的对称关系，并展示如何利用这些关系化简表达式、证明恒等式以及构造新的多项式方程。
1. The Big Idea: Connecting Roots and Coefficients | 核心思想：根与系数的联系
actorised form using its roots. If the roots are α, β, γ, ..., then comparing the expanded factorised form with the original coefficient form gives direct relationships between sums and products of roots and the coefficients. These are often called Vieta's formulas.
多项式方程如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀ = 0 可以利用其根写成因式分解形式。如果根为 α、β、γ、...，那么将展开后的因式分解形式与原始系数形式进行比较，就能直接得到根的和与积同系数之间的关系。这些关系通常称为韦达公式。
These relationships are symmetric: swapping any two roots does not change the value of the expression. Therefore α + β and αβ are symmetric functions. This symmetry is what makes them knowable from the coefficients alone.
这些关系是对称的：交换任意两个根不会改变表达式的值。因此 α + β 和 αβ 都是对称函数。正是这种对称性使得它们可以仅由系数确定。
2. Quadratic Polynomials: The Foundation | 二次多项式：基础
, with roots α and β, the factorised form is a(x − α)(x − β). Expanding gives ax² − a(α + β)x + aαβ. Comparing coefficients with ax² + bx + c immediately yields two key results.
对于二次方程 ax² + bx + c = 0，设其根为 α 和 β，因式分解形式为 a(x − α)(x − β)。展开得到 ax² − a(α + β)x + aαβ。与 ax² + bx + c 比较系数，立即得到两个关键结果。
α + β = −b/a
αβ = c/a
is the monic form. For example, if α + β = 5 and αβ = 6, the quadratic is x² − 5x + 6 = 0.
这两个恒等式让你能直接从系数求出根的和与积。它们还能让你在已知根的情况下重建二次方程，因为 x² − (根的和)x + 根的积 = 0 是首一形式。例如，若 α + β = 5 且 αβ = 6，则二次方程为 x² − 5x + 6 = 0。
3. Cubic Polynomials: Three Symmetric Sums | 三次多项式：三个对称和
with roots α, β and γ, the factorised form is a(x − α)(x − β)(x − γ). Expanding and comparing coefficients gives three central relationships involving the elementary symmetric sums.
对于三次方程 ax³ + bx² + cx + d = 0，设其根为 α、β 和 γ，因式分解形式为 a(x − α)(x − β)(x − γ)。展开并比较系数后，得到涉及基本对称和的三个核心关系。
α + β + γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a
Notice that the first sum has the opposite sign to b/a, the second sum of pairwise products has the same sign as c/a, and the triple product has the opposite sign to d/a. The signs alternate.
注意：第一个和式与 b/a 符号相反，两两乘积之和与 c/a 符号相同，三根之积与 d
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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