📚 Polynomial Functions: Graphs and Equations | IB数学：多项式函数的图像与方程
Polynomial functions are among the most studied objects in IB Mathematics. Their graphs reveal key features such as intercepts, turning points and end behaviour, while their equations can be solved using factorisation and the factor theorem. This article connects the algebraic and graphical sides of polynomials, providing a clear framework for exam success.
多项式函数是 IB 数学中最重要的研究对象之一。它的图像展现出截距、转折点和端点行为等关键特征，而它的方程可以通过因式分解和因式定理求解。本文将把多项式的代数与图像两方面联系起来，为考试成功提供一个清晰的框架。
1. Definition and General Form | 定义与一般形式
A polynomial function of degree n can be written in general form as
n 次多项式函数的一般形式可以写为
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, aₙ ≠ 0
Here n is a non-negative integer and all coefficients aₙ, aₙ₋₁, …, a₀ are real numbers. The highest power n is called the degree, and the coefficient aₙ is the leading coefficient. The term a₀ is the constant term.
这里 n 是非负整数，所有系数 aₙ, aₙ₋₁, …, a₀ 都是实数。最高次幂 n 称为次数，系数 aₙ 称为首项系数，项 a₀ 称为常数项。
A linear function is a degree-1 polynomial, a quadratic is degree-2, and a cubic is degree-3.
一次函数是次数为 1 的多项式，二次函数是次数为 2 的，三次函数是次数为 3 的。
2. Graphs and Key Features | 图像与关键特征
The graph of a polynomial is a smooth continuous curve with no breaks, corners or asymptotes.
多项式函数的图像是一条光滑连续曲线，没有间断、尖角或渐近线。
The y-intercept is found by evaluating f(0); thus it is always the constant term a₀.
y 截距通过计算 f(0) 得到，因此它总是常数项 a₀。
The x-intercepts are the real solutions of the equation f(x) = 0. A polynomial of degree n has at most n real roots.
x 截距是方程 f(x) = 0 的实数解。n 次多项式至多有 n 个实数根。
3. Factor Theorem and Roots | 因式定理与根
The factor theorem states that if f(a) = 0, then (x − a) is a factor of f(x). Conversely, if (x − a) is a factor, then a is a root.
因式定理指出：若 f(a) = 0，则 (x − a) 是 f(x) 的因式；反之，若 (x − a) 是因式，则 a 是根。
This allows us to factorise polynomials and reduce equations. For example, for f(x) = x³ − 7x + 6, since f(1) = 0, (x − 1) must be a factor.
这让我们可以分解多项式并化简方程。例如，对于 f(x) = x³ − 7x + 6，因为 f(1) = 0，所以 (x − 1) 必为因式。
4. Remainder Theorem | 余数定理
The remainder theorem gives the remainder when f(x) is divided by (x − a): it is simply f(a).
余数定理告诉我们，f(x) 除以 (x − a) 的余数就是 f(a)。
f(x) = (x − a)q(x) + f(a)
If f(a) = 0, the remainder is zero and the division is exact. It is often faster to substitute a than to perform long division.
如果 f(a) = 0，则余数为零，除法为整除。在求值时，代入 a 通常比长除法更快。
5. Repeated Roots and Multiplicity | 重根与重数
A root is repeated if a factor (x − r) appears more than once. The multiplicity is the exponent of the factor.
如果因式 (x − r) 出现多次，则根 r 是重根。重数就是该因式的指数。
If the multiplicity is odd, the graph crosses the x-axis at r; if it is even, the graph touches the x-axis and turns around without crossing.
如果重数为奇数，图像在 r 处穿过 x 轴；如果重数为偶数，图像在 r 处与 x 轴相切并折回，而不穿过。
The table below summarises the behaviour near x = r for different multiplicities.
下表总结了不同重数在 x = r 附近的行为。
| Multiplicity | Behaviour at x = r |
|---|---|
| 1 (odd) | Crosses straight through |
| 2 (even) | Touches and bounces |
| 3 (odd) | Crosses with a flattening at r |
Thus, multiplicity 1 and 3 lead to a crossing, while multiplicity 2 creates a ‘bounce’.
因此，重数 1 和 3 导致穿过，重数 2 产生“反弹”。
6. End Behaviour and Leading Coefficient | 端点行为与首项系数
For large |x|, the leading term aₙxⁿ dominates all other terms, so the end behaviour depends only on the degree and the sign of aₙ.
当 |x| 很大时，首项 aₙxⁿ 支配其余各项，所以端点行为只取决于次数和 aₙ 的符号。
For even degree, both ends of the graph go in the same direction; for odd degree, the ends go in opposite directions. This is often called the leading coefficient test.
对于偶次多项式，图像两端方向相同；对于奇次多项式，两端方向相反。这常称为首项系数判别法。
| Degree | Leading coefficient | End behaviour |
|---|---|---|
| Even | aₙ > 0 |
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