Polynomial Equations of Degree n | n次多项式方程

📚 Polynomial Equations of Degree n | n次多项式方程

Polynomial equations of degree n are fundamental in A-Level Mathematics. They appear in algebraic manipulation, curve sketching, and even in modelling real-world phenomena. Mastering these equations requires a clear understanding of factorisation, remainders, roots, and the relationship between coefficients and solutions.

n 次多项式方程是 A-Level 数学的基础内容。它们出现在代数运算、曲线绘图以及现实世界问题的建模中。掌握这类方程，需要清晰理解因式分解、余数、根，以及系数与解之间的关系。

1. Definition and Degree | 定义与次数

A polynomial of degree n has the general form P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where aₙ ≠ 0. The degree n is the highest power of x with a non-zero coefficient.

n 次多项式的一般形式为 P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀，其中 aₙ ≠ 0。次数 n 是系数不为零的 x 的最高幂次。

A polynomial equation of degree n is written as P(x) = 0. The solutions, or roots, are values of x that make P(x) equal to zero.

n 次多项式方程写作 P(x) = 0。解（即根）是使 P(x) 等于零的 x 值。

2. The Fundamental Theorem of Algebra | 代数基本定理

The Fundamental Theorem of Algebra states that a polynomial equation of degree n has exactly n roots, counting multiplicities, over the complex number system. This means that some roots may be repeated, and some may be non-real.

代数基本定理指出，在复数范围内，n 次多项式方程恰好有 n 个根（按重数计算）。这意味着某些根可能重复，某些根可能不是实数。

For example, (x − 1)² = 0 has only one distinct root, x = 1, but it is counted as two roots because of multiplicity 2.

例如，(x − 1)² = 0 只有一个不同的根 x = 1，但由于重数为 2，它被计作两个根。

3. The Remainder Theorem | 余数定理

The Remainder Theorem states that when a polynomial P(x) is divided by (x − a), the remainder is P(a). This is a powerful tool for evaluating polynomials without long division.

余数定理指出：当多项式 P(x) 除以 (x − a) 时，余数为 P(a)。这是一个无需长除法即可求多项式值的强大工具。

For instance, if P(x) = x³ − 2x² + 5x − 7, then P(2) = 8 − 8 + 10 − 7 = 3. Hence the remainder upon division by (x − 2) is 3.

例如，若 P(x) = x³ − 2x² + 5x − 7，则 P(2) = 8 − 8 + 10 − 7 = 3。因此除以 (x − 2) 的余数为 3。

4. The Factor Theorem | 因式定理

The Factor Theorem is a special case of the Remainder Theorem. It states that (x − a) is a factor of P(x) if and only if P(a) = 0. In other words, a is a root of the equation P(x) = 0.

因式定理是余数定理的特殊情形。它指出：(x − a) 是 P(x) 的因子当且仅当 P(a) = 0。换言之，a 是方程 P(x) = 0 的根。

This theorem allows us to factorise higher‑degree polynomials once we know one root. For example, if P(2) = 0 for P(x) = x³ − 4x² + x + 6, then (x − 2) is a factor, and polynomial division gives P(x) = (x − 2)(x² − 2x − 3).

该定理使我们在知道一个根后能够分解高次多项式。例如，若 P(x) = x³ − 4x² + x + 6 满足 P(2) = 0，则 (x − 2) 是因子，多项式除法得到 P(x) = (x − 2)(x² − 2x − 3)。

5. The Rational Root Theorem | 有理根定理

The Rational Root Theorem helps find rational roots of a polynomial with integer coefficients. Any rational root p/q (in lowest terms) must have p as a factor of the constant term a₀ and q as a factor of the leading coefficient aₙ.

有理根定理帮助寻找整系数多项式的有理根。任何最简形式的有理根 p/q 必须满足：p 是常数项 a₀ 的因子，q 是首项系数 aₙ 的因子。

For P(x) = 2x³ + 3x² − 8x + 3, possible rational roots are factors of 3 divided by factors of 2: ±1, ±3, ±1/2, ±3/2. Testing these values reveals x = 1 and x = −3/2 as roots.

对于 P(x) = 2x³ + 3x² − 8x + 3，可能的有理根为 3 的因子除以 2 的因子：±1、±3、±1/2、±3/2。逐一检验发现 x = 1 和 x = −3/2 是根。

6. Solving Cubic Equations | 解三次方程

Cubic equations are polynomial equations of degree 3. Standard strategies include finding one factor by trial (using the factor theorem), performing polynomial division, and then solving the resulting quadratic.

三次方程是次数为 3 的多项式方程。常用策略包括：通过试根（利用因式定理）找到一个因子，进行多项式除法，然后解得到的二次方程。

For example, solve x³ − 6x² + 11x − 6 = 0. Testing x = 1 gives 0, so (x − 1) is a factor. Dividing yields (x − 1)(x² − 5x + 6) = 0, and factorising the quadratic gives (x − 1)(x − 2)(x − 3) = 0. Thus the roots are x = 1, 2, 3.

例如，解 x³ − 6x² + 11x − 6 = 0。试 x = 1 得 0，所以 (x − 1) 是因子。除法得 (x − 1)(x² − 5x + 6) = 0，二次因式分解得 (x − 1)(x − 2)(x − 3) = 0。因此根为 x = 1, 2, 3。

7. Solving Quartic Equations | 解四次方程

Quartic equations have degree 4. A common approach is to factorise into two quadratics or into a product of linear factors if integer roots exist. Sometimes substitution can reduce the problem.

四次方程的次数为 4。常用方法是将其分解为两个二次因式，或在存在整数根时分解为线性因式的乘积。有时代换可以简化问题。

Consider x⁴ − 5x² + 4 = 0. Let y = x², giving y² − 5y + 4 = 0. This factors as (y − 1)(y − 4) = 0, so y = 1 or 4. Hence x² = 1 or x² = 4, giving x = ±1, ±2.

考虑 x⁴ − 5x² + 4 = 0。令 y = x²，得 y² − 5y + 4 = 0。因式分解为 (y − 1)(y − 4) = 0，所以 y = 1 或 4。因此 x² = 1 或 x² = 4，得到 x = ±1, ±2。

8. Complex Roots and Conjugate Pairs | 复数根与共轭对

If a polynomial has real coefficients, then any non‑real complex root must occur in a conjugate pair. That is, if a + bi is a root, then a − bi is also a root.

如果多项式具有实系数，那么任何非实复数根必定成共轭对出现。也就是说，若 a + bi 是根，则 a − bi 也是根。

For example, the quadratic equation x² + 2x + 5 = 0 has discriminant 4 − 20 = −16, so roots are x = −1 ± 2i. These form a conjugate pair.

例如，二次方程 x² + 2x + 5 = 0 的判别式为 4 − 20 = −16，因此根为 x = −1 ± 2i。它们构成共轭对。

Knowing one complex root of a higher‑degree polynomial lets you identify a quadratic factor: (x − (a + bi))(x − (a − bi)) = x² − 2ax + (a² + b²).

知道高次多项式的一个复数根，就能确定一个二次因子：(x − (a + bi))(x − (a − bi)) = x² − 2ax + (a² + b²)。

9. Vieta’s Formulas | 韦达定理

Vieta’s formulas relate the roots of a polynomial to its coefficients. For a cubic polynomial P(x) = ax³ + bx² + cx + d with roots α, β, γ, we have:

韦达定理将多项式的根与系数联系起来。对于三次多项式 P(x) = ax³ + bx² + cx + d，设其根为 α, β, γ，则有：

α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a

Similar relations exist for quartic and higher‑degree polynomials. These are useful when you know some roots and need to find others or to check your factorisation.

四次及更高次多项式也有类似关系。当已知部分根而需要求其他根，或需要检验因式分解是否正确时，韦达定理非常有用。

10. Using the Sum and Product of Roots | 利用根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum is α + β = −b/a and the product is αβ = c/a. This enables us to form a quadratic equation when given its roots.

对于二次方程 ax² + bx + c = 0，设根为 α 和 β，则根之和为 α + β = −b/a，根之积为 αβ = c/a。这使我们能够根据已知根构造二次方程。

If the roots are 3 and −2, then the equation is x² − (sum)x + product = 0, i.e. x² − x − 6 = 0.

若根为 3 和 −2，则方程为 x² − (和)x + 积 = 0，即 x² − x − 6 = 0。

11. Graphical Interpretation | 图像解释

The roots of P(x) = 0 correspond to the x‑intercepts of the graph y = P(x). A repeated root touches the x‑axis without crossing it. Non‑real roots produce no x‑intercepts.

方程 P(x) = 0 的根对应于图像 y = P(x) 与 x 轴的交点。重根处图像与 x 轴相切但不穿过；非实根不会产生 x 轴交点。

For a cubic with three distinct real roots, the graph crosses the x‑axis three times. If it has one real root and two complex conjugate roots, it crosses only once.

对于具有三个不同实根的三次方程，图像与 x 轴相交三次。若它有一个实根和两个共轭复根，则只相交一次。

12. Approximate Roots and the Intermediate Value Theorem | 近似根与介值定理

When exact roots are difficult to find, numerical methods can be used. The sign change rule (a consequence of the Intermediate Value Theorem) states that if P(a) and P(b) have opposite signs, then there is at least one root between a and b.

当精确根难以求得时，可以使用数值方法。变号规则（介值定理的推论）指出：若 P(a) 与 P(b) 异号，则在 a 与 b 之间至少存在一个根。

For example, for P(x) = x³ − 3x + 1, evaluating P(1) = −1 and P(2) = 3 shows a root lies between 1 and 2. Repeated interval bisection can refine this estimate.

例如，对 P(x) = x³ − 3x + 1，计算 P(1) = −1 和 P(2) = 3，可知一个根位于 1 与 2 之间。反复二分区间可以精化这个估计。

Mastery of polynomial equations of degree n is built on the factor and remainder theorems, understanding of complex roots, and facility with coefficient–root relationships. Practice with cubic and quartic factorisations, and always consider the Rational Root Theorem when hunting for first roots.

掌握 n 次多项式方程，关键在于熟练运用因式定理与余数定理、理解复数根，以及灵活处理系数与根的关系。多练习三次和四次方程的因式分解，并在寻找第一个根时始终考虑有理根定理。

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Polynomial Equations of Degree n | n次多项式方程