📚 Polynomial Equations of Degree n | n 次多项式方程
Polynomial equations are among the most important tools in A-Level Mathematics. A polynomial equation of degree n can model curves, optimise problems, and connect algebra with graph sketching. In this article, we will explore how to identify the degree, count roots, factorise efficiently, and solve cubic and quartic equations. We will also link the roots to coefficients and discuss common exam pitfalls.
多项式方程是 A-Level 数学中最重要的工具之一。一个 n 次多项式方程可以刻画曲线、解决优化问题，并将代数与函数图像联系起来。本文将探讨如何识别次数、计算根的个数、高效因式分解，以及求解三次和四次方程。我们还会把根与系数联系起来，并讨论常见考试陷阱。
1. What is a polynomial equation of degree n? | 什么是 n 次多项式方程？
A polynomial equation of degree n has the general form shown below. The number n is a positive integer, and the leading coefficient aₙ must be non-zero. If aₙ were zero, the degree would be lower than n.
一个 n 次多项式方程具有如下一般形式。数字 n 是正整数，首项系数 aₙ 必须不为零。如果 aₙ 为零，次数就会低于 n。
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0, aₙ ≠ 0
sible number of real roots and the maximum number of turning points on the graph y = p(x).
例如，2x³ − 5x + 1 = 0 是一个三次方程，而 x⁴ − 3x² + x − 7 = 0 是一个四次方程。次数 n 告诉我们实根的最大可能数量，以及图像 y = p(x) 上驻点的最大数量。
2. The Fundamental Theorem of Algebra and root counting | 代数基本定理与根的计数
The Fundamental Theorem of Algebra states that every non-zero single-variable polynomial of degree n has exactly n roots in the complex number system, provided each root is counted with its multiplicity. This does not mean there are always n distinct real roots.
代数基本定理指出，每一个非零的一元 n 次多项式在复数系中恰好有 n 个根，前提是每个根都按其重数计数。这并不意味着总是有 n 个不同的实根。
t no real roots. Its two roots are the complex numbers i and −i. In A-Level exam questions, you will often be asked to find all real roots, and sometimes to state how many roots the equation has.
在实数范围内，n 次多项式可以有少于 n 个实根。例如，x² + 1 = 0 的次数为 2，但没有实根。它的两个根是复数 i 和 −i。在 A-Level 考试题中，通常会要求你求出所有实根，有时还要求说明方程有多少个根。
3. The factor theorem and linear factors | 因式定理与线性因式
is a factor of p(x). Conversely, if (x − a) is a factor, then x = a is a root of p(x) = 0.
因式定理是根与因式之间的关键联系。如果 p(a) = 0，那么 (x − a) 是 p(x) 的一个因式。反过来，如果 (x − a) 是因式，那么 x = a 就是 p(x) = 0 的一个根。
If a degree n polynomial has n distinct roots α₁, α₂, ..., αₙ, then it can be written as a product of n linear factors. The leading coefficient aₙ must be included in front.
如果一个 n 次多项式有 n 个不同的根 α₁, α₂, ..., αₙ，那么它可以写成 n 个线性因式的乘积。首项系数 aₙ 必须放在前面。
p(x) = aₙ(x − α₁)(x − α₂) ... (x − αₙ)
This factorised form makes it clear why a degree n equation can have at most n real roots: each linear factor can contribute at most one real root, and there are exactly n linear factors in the full complex factorisation.
这种因式分解形式清楚地说明为什么 n 次方程最多有 n 个实根：每个线性因式最多贡献一个实根，而在完整的复数因式分解中恰好有 n 个线性因式。
4. The remainder theorem and polynomial division | 余数定理与多项式除法
The remainder theorem states that when a polynomial p(x) is divided by (x − a), the remainder is p(a). This is a quick way to test possible roots without doing full long division.
余数定理指出，当多项式 p(x) 除以 (x − a) 时，余数是 p(a)。这是快速检验可能根的方法，无需进行完整的长除法。
x − a) is a factor. Polynomial long division or synthetic division can then be used to reduce the degree of the polynomial. This process is essential when solving cubic and quartic equations.
如果 p(a) = 0，除法就是整除，且 (x − a) 是一个因式。然后可以使用多项式长除法或综合除法来降低多项式的次数。这个过程在求解三次和四次方程时至关重要。
p(x) ÷ (x − a) = q(x) with remainder r, where r = p(a)
1) gives the quotient x² − 5x + 6. This can then be factorised further as (x − 2)(x − 3).
例如，如果 p(x) = x³ − 6x² + 11x − 6 且 p(1) = 0，除以 (x − 1) 得到商 x² − 5x + 6。这可以进一步因式分解为 (x − 2)(x − 3)。
5. Solving cubic equations | 解三次方程
l root. In A-Level problems, the root is usually a small integer found by testing divisors of the constant term d.
要求解三次方程，例如 ax³ + bx² + cx + d = 0，首先要找到一个实根。在 A-Level 题目中，这个根通常是一个小整数，可以通过检验常数项 d 的因数来找到。
cubic by (x − r) to obtain a quadratic. Then solve the quadratic by factorisation, completing the square, or the quadratic formula. The three roots are r together with the two quadratic roots.
一旦找到一个根 x = r，就将三次式除以 (x − r) 得到一个二次式。然后通过因式分解、配方法或二次公式求解这个二次方程。三个根就是 r 以及二次方程的两个根。
x³ − 6x² + 11x − 6 = 0
Try x = 1: 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor.
Division gives x² − 5x + 6 = (x − 2)(x − 3).
Roots: x = 1, 2, 3.
Always write the final answer as a set of roots and check each one by substituting back into the original equation. This prevents sign errors and missed repeated roots.
始终将最终答案写为一组根，并将每个根代回原方程进行检验。这样可以防止符号错误和漏掉重根。
6. How many roots does a degree n equation have? | n 次方程有多少个根？
Over the complex numbers, a degree n equation has exactly n roots when multiplicities are counted. Over the real numbers, it can have fewer than n real roots. For example, a cubic must have at least one real root because its graph is continuous and goes to opposite infinities at each end.
在复数范围内，n 次方程在计算重数时恰好有 n 个根。在实数范围内，它的实根可能少于 n 个。例如，三次方程至少有一个实根，因为它的图像是连续的，并且在两端趋向于相反的无穷大。
If complex numbers are allowed, non-real roots always occur in conjugate pairs for polynomials with real coefficients. This means a real cubic can have either three real roots or one real root and two complex conjugate roots. It cannot have exactly two real roots and one non-real root.
如果允许使用复数，对于具有实系数的多项式，非实根总是以共轭对出现。这意味着实三次方程可以有三个实根，或者一个实根和两个共轭复数根。它不可能恰好有两个实根和一个非实根。
If p(x) has real coefficients and p(a + bi) = 0, then p(a − bi) = 0.
This conjugate pair property is very useful when one complex root is given and you need to find the remaining factors of a cubic or quartic equation.
当给出一个复数根并要求你找出三次或四次方程的其余因式时，这个共轭对性质非常有用。
7. Repeated roots | 重根
n x = a is a root of multiplicity k. For example, (x − 2)² = 0 has x = 2 as a double root.
当线性因式出现不止一次时，就会产生重根。如果 (x − a)ᵏ 是一个因式，那么 x = a 就是一个重数为 k 的根。例如，(x − 2)² = 0 中 x = 2 是一个二重根。
ltiplicity crosses the x-axis, while a root with even multiplicity touches the x-axis and turns back. This distinction is often tested in curve-sketching questions.
在图像 y = p(x) 上，奇重数的根会穿过 x 轴，而偶重数的根会接触 x 轴并折返。这种区别在曲线作图题中经常考查。
p(x) = (x + 1)²(x − 3)
x = −1 is a double root (touches the x-axis).
x = 3 is a simple root (crosses the x-axis).
. The equation x² − 4x + 4 = 0 has one distinct real root, x = 2, but it is a double root.
在计算根的个数时，始终要按重数计算重根。方程 x² − 4x + 4 = 0 有一个不同的实根 x = 2，但它是二重根。
8. Sum and product of roots | 根与系数的关系
The relationships between the roots and coefficients of a polynomial are often called Vieta's formulas. They are useful for checking answers, forming equations from given roots, and solving problems without finding every root explicitly.
多项式的根与系数之间的关系通常称为韦达定理。它们可用于检验答案、从给定根构造方程，以及在无需明确求出每个根的情况下解决问题。
| Quadratic ax² + bx + c = 0 | α + β = −b/a, αβ = c/a |
| Cubic ax³ + bx² + cx + d = 0 | α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a |
| Quartic ax⁴ + bx³ + cx² + dx + e = 0 | α + β + γ + δ = −b/a, αβ + αγ + αδ + βγ + βδ + γδ = c/a, αβγ + αβδ + αγδ + βγδ = −d/a, αβγδ = e/a |
5 and αβ = 6. This quickly confirms the roots are 2 and 3, since 2 + 3 = 5 and 2 × 3 = 6.
例如，如果 x² − 5x + 6 = 0 的根是 α 和 β，那么 α + β = 5，αβ = 6。这可以快速确认根是 2 和 3，因为 2 + 3 = 5 且 2 × 3 = 6。
9. Polynomial graphs and roots | 多项式图像与根
e n controls the overall shape. A polynomial of degree n has at most n − 1 turning points.
p(x) = 0 的根是图像 y = p(x) 的 x 轴截距。次数 n 决定了整体形状。n 次多项式最多有 n − 1 个驻点。
If n is even and aₙ is positive, both ends of the graph point upwards. If n is odd and aₙ is positive, the left end goes down and the right end goes up. The sign of aₙ reverses these directions if it is negative.
如果 n 是偶数且 aₙ 为正，图像两端都向上。如果 n 是奇数且 aₙ 为正，左端向下而右端向上。如果 aₙ 为负，这些方向会反转。
Even degree: y → ∞ as x → ±∞ when aₙ > 0.
Odd degree: y → −∞ as x → −∞ and y → ∞ as x → ∞ when aₙ > 0.
When sketching, mark the roots on the x-axis, note whether each root crosses or touches, plot the y-intercept p(0), and use the end behaviour to draw a smooth curve. You do not need to plot every point in the exam.
作图时，在 x 轴上标出根，注意每个根是穿过还是接触 x 轴，标出 y 轴截距 p(0)，并利用两端趋势绘制光滑曲线。考试中无需标出每一个点。
10. Worked example: solving a quartic equation | 例题：解四次方程
teger values that divide the constant term −6. The possible candidates are ±1, ±2, ±3, ±6.
我们来解四次方程 x⁴ − 5x³ + 5x² + 5x − 6 = 0。首先检验能整除常数项 −6 的小整数值。可能的候选数是 ±1, ±2, ±3, ±6。
p(1) = 1 − 5 + 5 + 5 − 6 = 0, so (x − 1) is a factor.
p(2) = 16 − 40 + 20 + 10 − 6 = 0, so (x − 2) is a factor.
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