📚 Polynomial Operations and Factorization Essentials | 多项式的运算与因式分解要点
This revision note covers the essential operations on polynomials and the main factorization techniques required in IB Mathematics. Mastery of these skills is vital for solving equations, sketching curves, and working with rational functions.
本文复习笔记涵盖 IB 数学中多项式的基本运算与主要因式分解技巧。掌握这些技能对于解方程、绘制曲线以及处理有理函数至关重要。
1. Standard Form and Basic Vocabulary | 标准形式与基本术语
A polynomial in the variable x is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ⋯ + a₁x + a₀, where n is a non-negative integer and the coefficients aᵢ are real numbers. The highest power n is called the degree; aₙ is the leading coefficient and a₀ is the constant term. If aₙ = 1, the polynomial is called monic.
变量 x 的多项式是形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ⋯ + a₁x + a₀ 的表达式，其中 n 是非负整数，系数 aᵢ 为实数。最高次数 n 称为次数；aₙ 为首项系数，a₀ 为常数项。若 aₙ = 1，则称该多项式为首一多项式。
For example, 4x³ – 5x + 2 has degree 3, leading coefficient 4, and constant term 2. A constant polynomial such as 7 has degree 0.
例如，4x³ – 5x + 2 的次数为 3，首项系数为 4，常数项为 2。常数多项式（如 7）的次数为 0。
2. Addition and Subtraction | 加法与减法
To add or subtract polynomials, combine like terms, that is, terms with the same power of x. It is usually helpful to write each polynomial in descending powers of x before performing the operation.
多项式的加法与减法通过合并同类项来完成，即合并具有相同 x 次幂的项。通常先将每个多项式按 x 的降幂排列，再进行运算会更有帮助。
Example: (2x³ + 3x² – x + 5) + (x³ – 2x² + 4x – 1) = (2x³ + x³) + (3x² – 2x²) + (-x + 4x) + (5 – 1) = 3x³ + x² + 3x + 4.
示例：(2x³ + 3x² – x + 5) + (x³ – 2x² + 4x – 1) = (2x³ + x³) + (3x² – 2x²) + (-x + 4x) + (5 – 1) = 3x³ + x² + 3x + 4。
For subtraction, remember to change the sign of every term in the polynomial being subtracted. For instance, (2x³ + 3x² – x + 5) – (x³ – 2x² + 4x – 1) = x³ + 5x² – 5x + 6.
做减法时，要改变被减多项式中每一项的符号。例如，(2x³ + 3x² – x + 5) – (x³ – 2x² + 4x – 1) = x³ + 5x² – 5x + 6。
3. Multiplication | 乘法
Polynomial multiplication is based on the distributive law: each term in the first polynomial must be multiplied by each term in the second polynomial. When multiplying powers of x, keep the base and add the exponents.
多项式乘法基于乘法分配律：第一个多项式中的每一项都必须乘以第二个多项式中的每一项。在计算 x 的幂相乘时，底数不变、指数相加。
Example: (x + 2)(x² – 3x + 1) = x³ – 3x² + x + 2x² – 6x + 2 = x³ – x² – 5x + 2.
示例：(x + 2)(x² – 3x + 1) = x³ – 3x² + x + 2x² – 6x + 2 = x³ – x² – 5x + 2。
The following special products appear frequently in IB questions:
以下特殊乘积公式在 IB 考试中经常出现：
(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
(a + b)(a – b) = a² – b²
These identities are equally useful in both expansion and factorization directions.
这些恒等式在展开和因式分解两个方向上都同样重要。
4. Polynomial Long Division | 多项式长除法
Polynomial long division is similar to ordinary number division. To divide P(x) by D(x), compare the leading terms, multiply the divisor by the appropriate term, subtract, and bring down the next term. Repeat until the remainder has degree less than the divisor.
多项式长除法与普通数的除法类似。用 D(x) 去除 P(x) 时，先比较首项，用适当项乘以除数，作差，再拖下下一项。重复这一过程，直到余式的次数低于除式的次数。
The result can be written as P(x) = D(x) × Q(x) + R(x), where Q(x) is the quotient and R(x) is the remainder.
结果可写为 P(x) = D(x) × Q(x) + R(x)，其中 Q(x) 是商式，R(x) 是余式。
Example: divide x³ – 2x² + 4x – 3 by x – 1. The quotient is x² – x + 3 and the remainder is 0, because x³ – 2x² + 4x – 3 = (x – 1)(x² – x + 3) + 0.
示例：用 x – 1 去除 x³ – 2x² + 4x – 3。商式为 x² – x + 3，余式为 0，因为 x³ – 2x² + 4x – 3 = (x – 1)(x² – x + 3) + 0。
5. Synthetic Division | 综合除法
When the divisor is a linear expression of the form x – c, synthetic division gives a compact alternative to long division. Write only the coefficients of P(x), bring down the first coefficient, multiply by c, add to the next coefficient, and continue until the last row is finished.
当除式为 x – c 这样的一次式时，综合除法是长除法的紧凑替代方法。只需写出 P(x) 的系数，把第一个系数拖下，乘以 c，加到下一个系数上，依此类推，直到完成最后一行。
For x³ – 2x² + 4x – 3 with divisor x – 1, c = 1. Using the coefficients 1, -2, 4, -3:
对于 x³ – 2x² + 4x – 3 除以 x – 1，c = 1。取系数 1, -2, 4, -3：
Bring down 1; multiply by 1 to get 1; add to -2 to get -1. Multiply -1 by 1 to get -1; add to 4 to get 3. Multiply 3 by 1 to get 3; add to -3 to get 0. The quotient is x² – x + 3 and the remainder is 0.
拖下 1；乘以 1 得 1；加到 -2 得 -1。将 -1 乘以 1 得 -1；加到 4 得 3。将 3 乘以 1 得 3；加到 -3 得 0。商式为 x² – x + 3，余式为 0。
Remember that for a divisor x + c, the synthetic division value is c with the opposite sign, namely -c.
注意：如果除式为 x + c，综合除法代入的数值应为 c 的相反数，即 -c。
6. Remainder Theorem | 余数定理
The remainder theorem states that when a polynomial P(x) is divided by x – c, the remainder is P(c). This follows immediately from P(x) = (x – c)Q(x) + R: setting x = c gives P(c) = R.
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