Algebraic Fractions | 代数分式

📚 Algebraic Fractions | 代数分式

Algebraic fractions are rational expressions of the form P(x)/Q(x), where P and Q are polynomials and Q is not identically zero. Mastering them is essential for A-level calculus, equation solving, and partial fractions.

代数分式是形如 P(x)/Q(x) 的有理式，其中 P 和 Q 是多项式，且 Q 不恒等于零。掌握代数分式对 A-level 微积分、方程求解和部分分式都至关重要。

1. Understanding Algebraic Fractions | 理解代数分式

An algebraic fraction is a quotient of two polynomials, such as (x² − 3x + 2)/(x − 1). The denominator cannot be zero, so every algebraic fraction carries a hidden domain restriction.

代数分式是两个多项式的商，例如 (x² − 3x + 2)/(x − 1)。分母不能为零，因此每个代数分式都带有隐藏的定义域限制。

Before simplifying, state excluded values. For example, in (x² − 3x + 2)/(x − 1), x = 1 makes the denominator zero, so x ≠ 1. After cancelling, the simplified expression x − 2 also requires x ≠ 1 even though x = 1 would not make the simplified denominator zero.

在化简之前，应先写出需要排除的值。例如，在 (x² − 3x + 2)/(x − 1) 中，x = 1 会使分母为零，所以 x ≠ 1。约分后得到的 x − 2 仍须注明 x ≠ 1，尽管 x = 1 不会使化简后的分母为零。

You should treat an algebraic fraction as a single rational expression only when its numerator and denominator are fully factorised; otherwise apparent cancellations may be invalid.

只有当分子和分母都完全因式分解后，才能把代数分式当作一个有理式进行约分；否则看似成立的约分可能是非法的。

2. Simplifying by Factorising | 因式分解化简

To simplify, factorise numerator and denominator completely, then cancel common polynomial factors. Example:

化简时，应先将分子和分母完全因式分解，再约去共同的整式因子。例如：

(x² + 5x + 6)/(x² − 4) = ((x + 2)(x + 3))/((x + 2)(x − 2)) = (x + 3)/(x − 2), x ≠ −2, 2

Notice that the factor x + 2 is cancelled, but the restriction x ≠ −2 must remain because it comes from the original denominator.

注意 x + 2 被约去，但限制条件 x ≠ −2 必须保留，因为它来自原分母。

Do not cancel individual terms inside a sum. For instance, (x + 2)/(x + 3) cannot be simplified to 2/3, and (x² + 1)/(x + 1) cannot be reduced by removing x from both numerator and denominator.

不要约去和式中的单个项。例如 (x + 2)/(x + 3) 不能化简为 2/3，(x² + 1)/(x + 1) 也不能通过消去分子和分母中的 x 来化简。

3. Multiplying and Dividing | 乘法与除法

For multiplication, factorise all numerators and denominators first, cancel common factors, then multiply across. For division, multiply by the reciprocal of the divisor.

乘法运算时，先将所有分子和分母因式分解，约去公因式，然后相乘；除法运算则乘以除式的倒数。

(x² − 9)/(x + 4) × (x + 4)/(x − 3) = ((x − 3)(x + 3))/(x + 4) × (x + 4)/(