Relationship Between the Roots of a Cubic Equation and Its Coefficients | 三次方程根与系数的关系

📚 Relationship Between the Roots of a Cubic Equation and Its Coefficients | 三次方程根与系数的关系

For any cubic equation, there is a beautiful and useful connection between its roots and its coefficients. These relationships, sometimes called Vieta’s formulas for cubics, let you calculate sums and products of the roots without solving the equation at all.

对于任意三次方程，其根与系数之间存在着优美而实用的联系。这些关系有时被称为“韦达定理”的三次形式，它让你无需解方程即可计算根的和与积。

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1. Why This Relationship Matters | 为什么这个关系很重要

In AQA A-level Mathematics, questions often ask you to find the sum, the sum of the product in pairs, or the product of the roots of a cubic equation. These values can be read directly from the coefficients, which saves time and reduces the risk of error.

在 AQA A-level 数学考试中，题目常常要求你求三次方程根的和、两根乘积之和或根的乘积。这些数值可以直接从系数中读出，这既节省时间，又能降低出错风险。

Beyond direct evaluation, these relationships allow you to form a new cubic when the roots are transformed, and to evaluate symmetric expressions such as α² + β² + γ² without finding α, β and γ explicitly.

除了直接计算之外，这些关系还允许你在根经过变换后构造新的三次方程，并计算诸如 α² + β² + γ² 这样的对称式，而不必显式求出 α、β 和 γ。

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2. Standard Form and Notation | 标准形式与记号

A cubic equation is usually written as:

ax³ + bx² + cx + d = 0, with a ≠ 0

We denote the three roots by α, β and γ. It is important to remember that a cubic equation has three roots, but some may be repeated or non-real. In AQA A-level work, we normally count roots with multiplicity.

三次方程通常写作：

ax³ + bx² + cx + d = 0，其中 a ≠ 0

我们用 α、β 和 γ 表示三个根。请记住，三次方程有三个根，但其中可能有重根或非实数根。在 AQA A-level 中，我们通常按重数计算根。

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3. The Three Key Relationships | 三个核心关系

If α, β and γ are the roots of ax³ + bx² + cx + d = 0, then the following three relationships hold:

如果 α、β 和 γ 是方程 ax³ + bx² + cx + d = 0 的根，那么以下三个关系成立：

Expression / 表达式	Formula / 公式
Sum of roots / 根之和	α + β + γ = −b/a
Sum of products in pairs / 两根乘积之和	αβ + αγ + βγ = c/a
Product of roots / 根之积	αβγ = −d/a

For a monic cubic, that is one of the form x³ + px² + qx + r = 0, the formulas become simpler:

对于首项系数为 1 的三次方程，即形如 x³ + px² + qx + r = 0 的方程，公式更简单：

Expression / 表达式	Formula / 公式
Sum of roots / 根之和	α + β + γ = −p
Sum of products in pairs / 两根乘积之和	αβ + αγ + βγ = q
Product of roots / 根之积	αβγ = −r

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4. Derivation From Factorised Form | 从因式形式推导

Suppose the leading coefficient is a and the roots are α, β and γ. Then the cubic can be written as:

假设首项系数为 a，根为 α、β 和 γ。那么这个三次方程可以写成：

a(x − α)(x − β)(x − γ) = 0

Expanding the factorised form gives:

展开因式形式得到：

(x − α)(x − β)(x − γ) = x³ − (α + β + γ)x² + (αβ + αγ + βγ)x − αβγ

Multiplying by a, we obtain:

乘以 a 后，我们得到：

a x³ − a(α + β + γ)x² + a(αβ + αγ + βγ)x − aαβγ

Now compare the coefficients with ax³ + bx² + cx + d:

现在将这个结果与 ax³ + bx² + cx + d 的各项系数进行对比：

b = −a(α + β + γ), c = a(αβ + αγ + βγ), d = −aαβγ

Rearranging gives exactly the three key relationships from Section 3.

重新整理后就得到第 3 小节中的三个核心关系。

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5. Worked Example: Reading the Coefficients | 例题：从系数读取关系

Consider the cubic equation:

考虑三次方程：

2x³ − 12x² + 22x − 12 = 0

Here a = 2, b = −12, c = 22 and d = −12. The roots are α, β and γ. Then:

这里 a = 2，b = −12，c = 22，d = −12。设根为 α、β 和 γ。那么：

α + β + γ = −(−12)/2 = 6

αβ + αγ + βγ = 22/2 = 11

αβγ = −(−12)/2 = 6

Check: this cubic has roots 1, 2 and 3. Indeed 1 + 2 + 3 = 6, 1×2 + 1×3 + 2×3 = 11, and 1×2×3 = 6.

验证：该三次方程的根为 1、2、3。确实有 1 + 2 + 3 = 6，1×2 + 1×3 + 2×3 = 11，以及 1×2×3 = 6。

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6. Computing Symmetric Expressions | 计算对称式

The three basic relationships can be combined to find more complicated symmetric expressions.

三个基本关系可以组合起来，用来求更复杂的对称表达式。

Let S₁ = α + β + γ, S₂ = αβ + αγ + βγ and S₃ = αβγ.

设 S₁ = α + β + γ，S₂ = αβ + αγ + βγ，S₃ = αβγ。

Sum of squares:

平方和：

α² + β² + γ² = S₁² − 2S₂

Sum of reciprocals: if no root is zero,

倒数之和：如果没有根为零，

1/α + 1/β + 1/γ = S₂/S₃

Sum of cubes:

立方和：

α³ + β³ + γ³ = S₁³ − 3S₁S₂ + 3S₃

For the worked example in Section 5, S₁ = 6, S₂ = 11 and S₃ = 6, so:

对于第 5 小节中的例题，S₁ = 6，S₂ = 11，S₃ = 6，因此：

α² + β² + γ² = 6² − 2×11 = 36 − 22 = 14

1/α + 1/β + 1/γ = 11/6

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7. Forming a Cubic With Transformed Roots | 构造根变换后的新三次方程

The root-coefficient relationships also allow us to build a new cubic whose roots are simple functions of the original roots.

根与系数的关系还允许我们构造一个新的三次方程，使其根是原根的一些简单函数。

Using the same cubic 2x³ − 12x² + 22x − 12 = 0, suppose we want a cubic with roots 2α, 2β and 2γ.

仍然使用方程 2x³ − 12x² + 22x − 12 = 0，假设我们想构造一个根为 2α、2β 和 2γ 的三次方程。

For the new roots:

对于新根：

Sum = 2α + 2β + 2γ = 2S₁ = 2×6 = 12

Pairwise sum = (2α)(2β) + (2α)(2γ) + (2β)(2γ) = 4S₂ = 4×11 = 44

Product = (2α)(2β)(2γ) = 8S₃ = 8×6 = 48

For a monic cubic x³ + px² + qx + r = 0 with these roots, we need:

对于首项系数为 1 的三次方程 x³ + px² + qx + r = 0，若要以这些数为根，需要：

p = −12, q = 44, r = −48

Therefore the transformed cubic is:

因此变换后的三次方程为：

x³ − 12x² + 44x − 48 = 0

Indeed the original roots are 1, 2 and 3, so the new roots are 2, 4

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