📚 6A Graphs and equations of polynomial functions | 6A 多项式函数的图像与方程
Polynomial functions form a fundamental building block in IB Mathematics. Understanding their graphs and equations is essential for solving a wide range of problems, from simple curve sketching to complex modeling. This article provides a comprehensive review of polynomial functions, their key features, algebraic techniques, and graphical behavior, aligning with the IB syllabus.
多项式函数是 IB 数学中的基础模块。理解它们的图像与方程对于求解从简单曲线绘制到复杂建模的各种问题至关重要。本文将对多项式函数进行全面回顾，涵盖其关键特征、代数技巧和图像行为，与 IB 课程大纲紧密对接。
1. Definition and Degree of Polynomial Functions | 多项式函数的定义与次数
A polynomial function has the form f(x) = an xn + an-1 xn-1 + … + a1 x + a0, where n is a non-negative integer, the coefficients ai are real constants, and an ≠ 0. The highest power n is called the degree of the polynomial. A constant function has degree 0 (except the zero function), a linear function has degree 1, a quadratic degree 2, and so on.
多项式函数的形式为 f(x) = an xn + an-1 xn-1 + … + a1 x + a0，其中 n 是非负整数，系数 ai 为实数常量，且 an ≠ 0。最高次幂 n 称为多项式的次数。常数函数（除零函数外）次数为 0，一次函数次数为 1，二次函数次数为 2，依此类推。
The degree determines the maximum number of turning points and the overall shape of the graph. A polynomial of degree d has at most d – 1 turning points and up to d real roots. For example, f(x) = 4x3 – 2x + 7 has degree 3, so it can have up to 2 turning points and up to 3 real roots.
次数决定了图像的最大拐点数和整体形状。d 次多项式最多有 d – 1 个拐点，至多有 d 个实根。例如，f(x) = 4x3 – 2x + 7 的次数为 3，因此它可以有至多 2 个拐点和至多 3 个实根。
2. Standard Form and Leading Coefficient | 标准形式与首项系数
The standard form arranges terms in descending powers of x. The coefficient an is the leading coefficient. It influences the end behavior and the steepness of the graph. If the leading coefficient is positive, the graph eventually rises to the right; if negative, it falls to the right. For example, for f(x) = -2x4 + x2 – 5, the leading coefficient is -2.
标准形式按 x 的降幂排列各项。系数 an 为首项系数。它影响末尾行为和图像的陡峭程度。若首项系数为正，图像最终向右上升；若为负，则向右下降。例如，对于 f(x) = -2x4 + x2 – 5，首项系数为 -2。
When a polynomial is not in standard form, always rearrange it before analyzing its features. The degree and leading coefficient are then easily identified. In IB exam questions, you may be given a factored form and asked to determine these properties without fully expanding.
当多项式不是标准形式时，务必先将其重排，再分析其特征。这样次数和首项系数便一目了然。在 IB 考试中，你可能会遇到给定因式形式的问题，要求在不完全展开的情况下判断这些性质。
3. End Behavior of Polynomials | 多项式图像的末尾行为
The end behavior describes how the graph behaves as x → +∞ and x → -∞. It depends entirely on the degree (even or odd) and the sign of the leading coefficient. For even degrees, both ends go in the same direction; for odd degrees, they go in opposite directions.
末尾行为描述了当 x → +∞ 和 x → -∞ 时图像的表现。它完全取决于次数（奇或偶）和首项系数的符号。对于偶次多项式，两端走向相同；对于奇次多项式，两端走向相反。
| Degree (次数) | Leading Coefficient (首项系数) | End Behavior (末尾行为) |
|---|---|---|
| Even (偶) | Positive (正) | Up on both ends (两端向上) |
| Even (偶) | Negative (负) | Down on both ends (两端向下) |
| Odd (奇) | Positive (正) | Down on left, Up on right (左端向下，右端向上) |
| Odd (奇) | Negative (负) | Up on left, Down on right (左端向上，右端向下) |
Memorising this pattern is crucial for sketching graphs quickly. You can always verify with a few test points or by considering the dominant term an xn when |x| is large.
牢记此规律对快速绘制图像至关重要。你总可以通过代入几个测试值，或考虑 |x| 很大时的主导项 an xn 来加以验证。
4. Roots, Zeros, and X-intercepts | 根、零点与 x 轴截距
The solutions to f(x)=0 are called roots or zeros of the polynomial. Graphically, they are the x-intercepts where the curve meets or touches the x-axis. Real roots correspond to x-intercepts, while complex roots occur in conjugate pairs and do not appear as intercepts on the real plane.
方程 f(x)=0 的解称为多项式的根或零点。在图像上，它们是对应曲线与 x 轴相交或相切的 x 截距。实根对应 x 轴截距，而复数根以共轭对出现，在实数平面上不是截距。
A polynomial of degree d has exactly d roots in the complex number system (Fundamental Theorem of Algebra), but the number of real roots can be less. For example, x2 + 1 = 0 has no real roots, so its graph never crosses the x-axis. The maximum number of real roots equals the degree.
根据代数基本定理，d 次多项式在复数域中有恰好 d 个根，但实根的数量可能更少。比如，x2 + 1 = 0 无实根，其图像永不穿过 x 轴。实根的最大数目等于次数。
5. Factor Theorem and Remainder Theorem | 因式定理与余数定理
The Remainder Theorem states that when a polynomial f(x) is divided by (x – c), the remainder is f(c). The Factor Theorem is a direct consequence: if f(c) = 0, then (x – c) is a factor of f(x). These theorems are powerful tools for factoring polynomials and finding zeros systematically.
余数定理指出，当多项式 f(x) 除以 (x – c) 时，余数为 f(c)。因式定理是其直接推论：如果 f(c) = 0，则 (x – c) 是 f(x) 的因式。这些定理是系统地进行因式分解和寻找零点的有力工具。
For example, consider f(x) = x3 – 4x2 + x + 6. Testing c = 2 gives f(2) = 8 – 16 + 2 + 6 = 0, so (x – 2) is a factor. Similarly, testing c = -1 yields f(-1) = -1 – 4 – 1 + 6 = 0, so (x + 1) is also a factor. With these, the polynomial can be factored further.
例如，考虑 f(x) = x3 – 4x2 + x + 6。检验 c = 2 得 f(2) = 8 – 16 + 2 + 6 = 0，故 (x – 2) 是因式。类似地，检验 c = -1 得 f(-1) = -1 – 4 – 1 + 6 = 0，故 (x + 1) 也是因式。借助这些，可进一步分解该多项式。
6. Synthetic Division and Finding Factors | 综合除法与寻找因式
Synthetic division provides a streamlined way to divide a polynomial by a linear factor of the form (x – c). It works by using only the coefficients and performing repeated multiply-add steps. Combined with the Factor Theorem, it enables efficient factorization of higher-degree polynomials.
综合除法提供了一个多项式除以 (x – c) 形式线性因式的简化方法。它仅使用系数并执行重复的乘加步骤。与因式定理结合，可高效地对高次
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