Binomial Expansion | 二项式展开

📚 Binomial Expansion | 二项式展开

Binomial expansion is a core algebra topic in Edexcel A Level Mathematics. It allows you to expand expressions of the form (a+b)ⁿ without multiplying out each bracket by hand. Understanding the binomial coefficients, the general term, and the validity conditions for fractional or negative powers is essential for both Pure Mathematics and applied questions.

二项式展开是爱德思 A-Level 数学纯数部分的核心代数内容。它可以让你在不逐项展开括号的情况下，直接展开形如 (a+b)ⁿ 的表达式。掌握二项式系数、通项以及分数或负指数展开的有效性条件，对纯数和应用题都非常重要。

1. Pascal’s Triangle | 杨辉三角

For small positive integer powers, Pascal’s triangle gives a quick way to find coefficients. Row n of the triangle corresponds to the coefficients in the expansion of (a+b)ⁿ. For example, row 4 is 1 4 6 4 1, so (a+b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.

对于较小的正整数指数，杨辉三角是快速找到系数的工具。三角形第 n 行对应 (a+b)ⁿ 展开式中的系数。例如第 4 行是 1 4 6 4 1，因此 (a+b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴。

Each entry in Pascal’s triangle is the sum of the two entries directly above it. This pattern gives a geometric and recursive view of binomial coefficients, but it becomes inefficient for large n.

杨辉三角中每个数等于其上方两个数之和。这种规律从几何和递推角度解释了二项式系数，但当 n 较大时，这种方法效率不高。

2. Factorials and nCr | 阶乘与二项式系数

The binomial coefficient nCr, also written as C(n, r), counts the number of ways to choose r objects from n objects without regard to order. It is defined by nCr = n! / [r!(n-r)!], where n! = n × (n-1) × … × 1.

二项式系数 nCr，也写作 C(n, r)，表示从 n 个不同对象中无序选出 r 个对象的组合数。其定义为 nCr = n! / [r!(n-r)!]，其中 n! = n × (n-1) × … × 1。

For example, 5C2 = 5! / (2!3!) = (5 × 4 × 3 × 2 × 1) / [(2 × 1)(3 × 2 × 1)] = 10. You must be able to evaluate such coefficients efficiently and understand the symmetry property nCr = nC(n-r).

例如 5C2 = 5! / (2!3!) = (5 × 4 × 3 × 2 × 1) / [(2 × 1)(3 × 2 × 1)] = 10。你必须能快速计算这类系数，并理解对称性 nCr = nC(n-r)。

3. Binomial Theorem for Positive Integer n | 正整数指数二项式定理

For any positive integer n, the binomial theorem states that (a+b)ⁿ = Σ from r=0 to n of nCr × aⁿ⁻ʳ × bʳ. The terms start with r = 0 and end with r = n, giving n + 1 terms in total.

对于任意正整数 n，二项式定理指出 (a+b)ⁿ = Σ（r 从 0 到 n）nCr × aⁿ⁻ʳ × bʳ。展开项从 r = 0 开始到 r = n 结束，共有 n + 1 项。

(a+b)ⁿ = nC0 aⁿ + nC1 aⁿ⁻¹ b + nC2 aⁿ⁻² b² + … + nCn bⁿ

Note the pattern: the powers of a decrease from n to 0, while the powers of b increase from 0 to n. The coefficients are exactly the entries of row n in Pascal’s triangle.

注意规律：a 的指数从 n 逐项下降到 0，而 b 的指数从 0 逐项上升到 n。系数正是杨辉三角第 n 行的数字。

4. Expanding (a+b)ⁿ in Practice | 实际问题中的 (a+b)ⁿ 展开

In Edexcel exam questions, you often need to expand expressions like (2x – 3)⁵. You can set a = 2x and b = -3, then apply the binomial theorem term by term.

在爱德思考试题中，你经常需要展开如 (2x – 3)⁵ 这样的表达式。可以令 a = 2x，b = -3，然后逐项应用二项式定理。

For example, (2x – 3)⁵ = 5C0(2x)⁵ + 5C1(2x)⁴(-3) + 5C2(2x)³(-3)² + 5C3(2x)²(-3)³ + 5C4(2x)(-3)⁴ + 5C5(-3)⁵. Simplify each term to obtain 32x⁵ – 240x⁴ + 720x³ – 1080x² + 810x – 243.

例如 (2x – 3)⁵ = 5C0(2x)⁵ + 5C1(2x)⁴(-3) + 5C2(2x)³(-3)² + 5C3(2x)²(-3)³ + 5C4(2x)(-3)⁴ + 5C5(-3)⁵。化简每一项得到 32x⁵ – 240x⁴ + 720x³ – 1080x² + 810x – 243。

Always simplify fully, including signs and powers of constants. In the exam, marks are often awarded for correct binomial coefficients and simplifying powers correctly.

一定要完整化简，包括符号和常数的幂。考试中，正确的二项式系数和正确化简各次幂通常都有给分。

5. Finding a Particular Term | 求特定项

A very common question asks you to find a specific term, such as the coefficient of x⁶, without writing out the whole expansion. You can use the general term formula Tr+1 = nCr × aⁿ⁻ʳ × bʳ.

一个非常常见的题型是求特定项，例如 x⁶ 的系数，而不需要写出整个展开式。你可以使用通项公式 Tr+1 = nCr × aⁿ⁻ʳ × bʳ。

For example, to find the coefficient of x⁶ in (2 + x)¹⁰, set a = 2, b = x, n = 10. The general term is 10Cr × 2¹⁰⁻ʳ × xʳ. We need r = 6, so the term is 10C6 × 2⁴ × x⁶ = 210 × 16 × x⁶ = 3360x⁶, so the coefficient is 3360.

例如，求 (2 + x)¹⁰ 中 x⁶ 的系数，令 a = 2，b = x，n = 10。通项为 10Cr × 2¹⁰⁻ʳ × xʳ。我们需要 r = 6，所以该项为 10C6 × 2⁴ × x⁶ = 210 × 16 × x⁶ = 3360x⁶，因此系数为 3360。

Be careful with questions that ask for the term independent of x or for two terms whose coefficients are equal. In such cases, set the power of x equal to zero or equate coefficients.

注意题目可能会问不含 x 的项，或者两项系数相等的情况。此时应令 x 的指数等于 0，或令两个系数相等。

6. Binomial Expansion for Rational and Negative Powers | 有理数与负指数二项式展开

When n is not a positive integer, the finite binomial theorem no longer applies. Instead, for any rational n and |x| < 1, the expansion is infinite: (1+x)ⁿ = 1 + nx + [n(n-1)/2!] x² + [n(n-1)(n-2)/3!] x³ + ...

当 n 不是正整数时，有限项二项式定理不再适用。相反，对于任意有理数 n 且 |x| < 1，展开式为无穷级数：(1+x)ⁿ = 1 + nx + [n(n-1)/2!] x² + [n(n-1)(n-2)/3!] x³ + ...

This formula is given in the Edexcel formula booklet, but you must know how to use it correctly. The expansion continues forever because n is not

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Binomial Expansion | 二项式展开