📚 Polynomial Equations | 多项式方程
Polynomial equations form a fundamental pillar of algebra, connecting curve sketching, complex numbers, and advanced calculus. In IB Mathematics, mastering polynomial equations means being able to solve equations up to degree four, use the factor and remainder theorems fluently, and interpret the geometric meaning of roots. This article systematically covers the key concepts required for success in the IB examination.
多项式方程是代数的重要基石，将曲线作图、复数与高等微积分紧密联系起来。在IB数学中，掌握多项式方程意味着能够熟练求解四次及以下方程，灵活运用因式定理和余式定理，并能从几何角度解读根的意义。本文系统地梳理了IB考试所要求的关键概念。
1. Definition and Standard Form | 定义与标准形式
riable x is an equation of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0, where n is a non‑ ents a₀, a₁, ..., aₙ are real or complex constants, and aₙ ≠ 0. The expression on the left‑hand side is called a polynomial, and the equation is said to be of degree n.
关于变量x的多项式方程是形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0 的方程，其中n为非负整数，系数 a₀, a₁, ..., aₙ 为实数或复数常数，且 aₙ ≠ 0。左侧的表达式称为多项式，该方程的次数为n。
4 = 0 should be rearranged as −x³ + 2x + 4 = 0 or x³ − 2x − 4 = 0 after multiplying by −1.
将多项式写成标准形式（按x的降幂排列）便于识别次数和首项系数。例如，3x⁴ − 5x² + 2x − 7 = 0 已是标准形式，而 2x − x³ + 4 = 0 应整理为 −x³ + 2x + 4 = 0，或乘以 −1 后得到 x³ − 2x − 4 = 0。
2. Degree and Leading Coefficient | 次数与首项系数
The degree of a polynomial equation is the highest exponent of the variable. The coefficient of that highest‑power term is called the leading coefficient. These two attributes largely determine the end behaviour of the corresponding polynomial function and the maximum number of real roots.
多项式方程的次数是变量指数的最高值。该最高次项的系数称为首项系数。这两个属性在很大程度上决定了对应多项式函数的终端行为以及实数根的最大可能个数。
A key fact: a polynomial equation of degree n has exactly n roots in the complex number system (counting multiplicities). For instance, a cubic equation always has three roots, though some may be repeated or non‑real.
一个关键事实是：在复数系统中，n次多项式方程恰好有n个根（计重数）。例如，三次方程始终有三个根，尽管其中一些可能相同或为非实数。
3. Roots and Zeros | 根与零点
(x), the same value r is called a zero of the function, and graphically it represents an x‑intercept. Solving a polynomial equation therefore means finding all its roots.
多项式方程 P(x) = 0 的根是指满足 P(r) = 0 的数r。当我们考虑函数 y = P(x) 时，同样的值r被称为函数的零点，在图像上它表示与x轴的交点。因此，解多项式方程就是求出其所有的根。
The terms 'root' and 'zero' are often used interchangeably. However, in IB contexts, 'root' usually refers to the solution of an equation, while 'zero' refers to the x‑coordinate of a point where the graph meets the x‑axis.
“根”和“零点”这两个术语经常互换使用。然而，在IB语境中，“根”通常指方程的解，而“零点”指图像与x轴交点的x坐标。
4. Factor Theorem and Remainder Theorem | 因式定理与余式定理
− a), the remainder is P(a). If P(a) = 0, then (x − a) is a factor of P(x). Symbolically:
P(a) = 0 ⇔ (x − a) is a factor of P(x).
因式定理是余式定理的直接推论。余式定理指出：多项式 P(x) 除以 (x − a) 时，余式为 P(a)。若 P(a) = 0，则 (x − a) 是 P(x) 的一个因式。用符号表示：P(a) = 0 ⇔ (x − a) 是 P(x) 的因式。
) is a factor. The remaining quadratic factor can be found by long division or synthetic division.
这一定理为因式分解和方程求解提供了有力的工具。例如，求解 x³ − 4x² + x + 6 = 0 时，检验可能的整数根。P(2) = 8 − 16 + 2 + 6 = 0，所以 (x − 2) 是因式。剩余的二次因式可通过长除法或综合除法求得。
5. Rational Root Theorem | 有理根定理
y rational root p/q (in lowest terms) must have p dividing the constant term a₀ and q dividing the leading coefficient aₙ.
对于整系数多项式，有理根定理给出了一组有限个可能的有理根。若多项式 aₙxⁿ + ... + a₀ = 0 具有整系数，则任何有理根 p/q（最简分数）必须满足：p整除常数项 a₀，且 q 整除首项系数 aₙ。
₂, etc. Testing reveals x = 2 is a root. This theorem often dramatically reduces the trial‑and‑error effort.
考虑 2x³ − 3x² − 12x + 20 = 0。可能的有理根分子p：±1, ±2, ±4, ±5, ±10, ±20；分母q：±1, ±2。因此可能的根：±1, ±2, ±4, ±5, ±10, ±20, ±½, ±⁵⁄₂ 等。检验发现 x = 2 是一个根。这一定理通常能大幅减少试错的工作量。
6. Fundamental Theorem of Algebra | 代数基本定理
The Fundamental Theorem of Algebra guarantees that every non‑constant polynomial equation with complex coefficients has at least one complex root. As a corollary, a polynomial of degree n can be factored into n linear factors over the complex numbers. In other words, every degree‑n equation has exactly n complex roots (counting multiplicities).
代数基本定理保证，每个非常系数的复系数多项式方程至少有一个复数根。由此推论，一个n次多项式可以在复数域上分解为n个一次因式的乘积。换言之，每一个n次方程恰好有n个复数根（计重数）。
This theorem explains why a quadratic equation always has two roots (real or complex), a cubic has three roots, and so on. Even when some roots are not real, they still exist in the complex plane and can be handled with complex number arithmetic.
这一定理解释了为何二次方程始终有两个根（实的或复的），三次方程有三个根，依此类推。即使某些根并非实数，它们仍然存在于复平面中，并且可以利用复数运算来处理。
7. Complex Roots and Conjugate Pairs | 复数根与共轭对
If a polynomial has real coefficients, any non‑real complex roots must appear in conjugate pairs. That is, if a + bi (b ≠ 0) is a root, then its complex conjugate a − bi is also a root. This property is crucial when constructing polynomials from given roots or when finding all roots of a real equation.
若多项式具有实系数，则任何非实数复数根必然以共轭对的形式出现。即，若 a + bi（b ≠ 0）是根，那么它的共轭复数 a − bi 也是根。在根据给定根构造多项式或求取实系数方程的所有根时，这一性质至关重要。
For example, knowing that 2 + i is a root of a cubic with real coefficients immediately tells you that 2 − i is also a root. The third root must be real and can be determined by the fact that the sum of the roots equals −b/a.
例如，已知 2 + i 是一个实系数三次方程的根，便可立刻得知 2 − i 也是根。第三个根必定为实数，且可通过根的和等于 −b/a 这一关系来确定。
8. Relationship Between Roots and Coefficients | 根与系数的关系 (Vieta)
Vieta' oefficients of a polynomial. For a quadratic equation ax² + bx + c = 0 with roots α and β:
α + β = −b/a, αβ = c/a.
Vieta公式给出了多项式根与系数之间的直接联系。对于具有根α和β的二次方程 ax² + bx + c = 0，有：α + β = −b/a，αβ = c/a。
For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ:
α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.
对于具有根α, β, γ的三次方程 ax³ + bx² + cx + d = 0，有：α + β + γ = −b/a，αβ + βγ + γα = c/a，αβγ = −d/a。
These relations are extremely useful when one root is known and the others need to be found without heavy factorisation, or when constructing an equation from given conditions about the roots, such as “the sum of the roots is 5 and the product is 6”.
当已知一个根需要求出其他根而不想使用繁琐的因式分解时，或者当需要根据根的给定条件（例如“根之和为5，积为6”）构造方程时，这些关系极为有用。
9. Solving Polynomial Equations by Factoring | 通过因式分解解多项式方程
g difference of squares (x² − a² = (x − a)(x + a)), sum/difference of cubes, and quadratic‑form polynomials (e.g. x⁴ − 5x² + 4 = 0 treated as (x²)² − 5(x²) + 4 = 0).
因式分解仍是求解多项式方程最为直观的方法。常用的策略包括：提取公因式、分组分解、识别平方差 (x² − a² = (x − a)(x + a))、立方和／立方差，以及二次形式的多项式（例如将 x⁴ − 5x² + 4 = 0 视为 (x²)² − 5(x²) + 4 = 0 来处理）。
equal to zero yields all roots. Always check for repeated factors to account for multiplicity.
一旦多项式在实数范围内被完全分解为一次因式和不可约二次因式，零乘积性质指出：若 A × B = 0，则 A = 0 或 B = 0。令每一个因式等于零，即可求出所有的根。务必检查是否有重复因式，以考虑重数。
10. Synthetic Division | 综合除法
Synthetic division is a streamlined method of dividing a polynomial by a linear divisor of the form (x − c). It is particularly efficient when testing possible roots suggested by the Rational Root Theorem. The process uses only the coefficients and the value c, reducing the risk of algebraic errors compared with polynomial long division.
综合除法是一种将多项式除以形如 (x − c) 的一次除式的简捷方法。在检验由有理根定理提供的有理根时尤其高效。这一方法仅使用系数以及c的值，与多项式长除法相比，能减少代数出错的几率。
To divide 2x³ − 3x² − 12x + 20 by (x − 2), write the coefficients 2, −3, −12, 20. Bring down the 2, multiply by 2 (the root) to get 4, add to −3 to get 1, multiply by 2 to get 2, add to −12 to get −10, multiply by 2 to get −20, and add to 20 to get 0, confirming no remainder. The quotient is 2x² + x − 10.
以 (x − 2) 除以 2x³ − 3x² − 12x + 20 为例，写出系数 2, −3, −12, 20。将2下移，乘以2（作为根）得4，与−3相加得1；再乘以2得2，与−12相加得−10；再乘以2得−20，与20相加得0，确认余数为零。商式为 2x² + x − 10。
11. Graphs of Polynomial Functions and Root Behaviour | 多项式函数图像与根的行为
The graph of a polynomial function is a smooth, continuous curve. The multiplicity of a root determines the graph's behaviour at the x‑intercept. If a root has odd multiplicity (1, 3, 5, …), the graph crosses the x‑axis at that point. If a root has even multiplicity (2, 4, 6, …), the graph touches the x‑axis and turns around, staying on the same side.
多项式函数的图像是一条光滑、连续的曲线。根的重数决定了图像在x轴交点处的行为。若根的重数为奇数（1, 3, 5, …），图像在该点穿过x轴。若根的重数为偶数（2, 4, 6, …），图像接触x轴后折返，保持在x轴的同侧。
city 2 (touching). Understanding these patterns helps in sketching graphs without plotting numerous points and in interpreting the nature of the solutions.
例如，P(x) = (x + 1)³(x − 2)² 中，x = −1 是重数为3的根（穿过），而 x = 2 是重数为2的根（触碰）。理解这些模式有助于在无需描点的情况下概略绘制图像，并有助于解释解的特性。
12. Applications and Modelling | 应用与建模
Polynomial equations frequently model real‑ a cubic polynomial, and solving V(x) = constant reduces to solving a polynomial equation.
多项式方程经常用于对现实情景进行建模，如体积优化、轨迹以及经济函数等。一个典型的问题是：从矩形板材的四个角各剪去边长为x的正方形以制作无盖盒。体积 V(x) 是一个三次多项式，求解 V(x) = 常数 便归结为解一个多项式方程。
In such applications, domain restrictions (e.g. x > 0 and x < half the shorter side) and the interpretation of extraneous roots (negative or too large) are vital. IB exam questions often require setting up the polynomial model, finding relevant roots, and then interpreting them in the original context.
在此类应用中，定义域限制（例如 x > 0 且 x < 较短边长的一半）以及对增根（负根或过大的根）的解读至关重要。IB考题经常要求建立多项式模型，求出相关的根，然后在原始情境中对其进行解释。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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