Binomial Expansion for A-Level Edexcel Maths | A-Level Edexcel 数学二项展开式精讲

📚 Binomial Expansion for A-Level Edexcel Maths | A-Level Edexcel 数学二项展开式精讲

The binomial expansion is a core topic in Edexcel A-Level Mathematics. It allows you to expand expressions of the form (a+b)ⁿ without multiplying brackets directly. This skill is tested regularly in Pure Mathematics papers, and it also underpins later work in calculus, series, and approximation.

二项展开式是 Edexcel A-Level 数学中的核心考点。它帮助你展开形如 (a+b)ⁿ 的表达式，无需逐项相乘括号。这一技能在纯数学试卷中经常考查，也为后续微积分、级数和近似计算打下基础。

1. What is Binomial Expansion? | 什么是二项展开式？

A binomial is an algebraic expression with two terms, such as (2x+3) or (1−x). The binomial expansion gives a formula for raising a binomial to a positive integer power, producing a polynomial. For example, (x+y)² expands to x²+2xy+y², and (x+y)³ expands to x³+3x²y+3xy²+y³.

二项式是指含有两个项的代数式，例如 (2x+3) 或 (1−x)。二项展开式给出了将二项式进行正整数次幂运算的公式，结果是一个多项式。例如，(x+y)² 展开为 x²+2xy+y²，(x+y)³ 展开为 x³+3x²y+3xy²+y³。

Edexcel exam questions often ask you to expand a binomial up to a given term, or to find one specific coefficient. You must be able to use both Pascal’s triangle for small powers and the general binomial theorem for larger or non-integer powers.

Edexcel 考试题目经常要求你将某个二项式展开到指定项，或者求出某一个特定系数。你必须熟练掌握小指数时使用的帕斯卡三角形，以及处理较大指数或非整数指数时的一般二项式定理。

2. Pascal’s Triangle for Small Powers | 小指数的帕斯卡三角形

For small positive integer powers, Pascal’s triangle gives the coefficients quickly. The first few rows are:

对于较小的正整数次幂，帕斯卡三角形可以快速给出系数。前几行如下：

Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1

Each number is the sum of the two numbers directly above it. For example, in row 4, the 6 comes from 3+3 in row 3. To expand (x+y)⁴, you use row 4 coefficients and decreasing powers of x and increasing powers of y.

每个数字等于其上方两个数字之和。例如，第 4 行中的 6 来自第 3 行的 3+3。要展开 (x+y)⁴，你使用第 4 行的系数，并让 x 的指数依次减小、y 的指数依次增大。

(x+y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

3. The Binomial Theorem for (1+x)ⁿ | (1+x)ⁿ 的二项式定理

When n is a positive integer, the binomial theorem states that:

当 n 为正整数时，二项式定理指出：

(1+x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … + xⁿ

Here the coefficient of xʳ is nCr, which is read as “n choose r”. The general term is nCr xʳ. This formula is especially useful when n is large, because Pascal’s triangle becomes impractical.

这里 xʳ 的系数是 nCr，读作 “n 选 r”。通项公式为 nCr xʳ。当 n 很大时，这个公式特别有用，因为帕斯卡三角形变得不切实际。

For (1−x)ⁿ, simply replace x by −x. The signs will alternate because (−x)ʳ is positive for even r and negative for odd r.

对于 (1−x)ⁿ，只需将 x 替换为 −x。符号会交替出现，因为 (−x)ʳ 在 r 为偶数时为正，r 为奇数时为负。

4. General Binomial Expansion for (a+b)ⁿ | (a+b)ⁿ 的一般展开

To expand (a+b)ⁿ, where a and b are any terms and n is a positive integer, use the general formula:

要展开 (a+b)ⁿ，其中 a 和 b 是任意项且 n 为正整数，使用一般公式：

(a+b)ⁿ = Σₖ₌₀ⁿ nCk aⁿ⁻ᵏ bᵏ

This means you take the sum from k=0 to n, where each term has coefficient nCk, a raised to the power n−k, and b raised to the power k. The powers of a decrease while the powers of b increase.

这意味着从 k=0 到 n 求和，每一项的系数为 nCk，a 的指数为 n−k，b 的指数为 k。a 的幂逐步递减，b 的幂逐步递增。

For example, to expand (2x−3y)³, set a=2x and b=−3y, then apply the formula with n=3.

例如，要展开 (2x−3y)³，令 a=2x，b=−3y，然后代入 n=3 的公式。

(2x−3y)³ = 8x³ − 36x²y + 54xy² − 27y³

5. Factorial Notation and nCr | 阶乘符号与 nCr

The binomial coefficient nCr can be calculated using factorials:

二项式系数 nCr 可以用阶乘计算：

nCr = n! / [r!(n−r)!]

Here n! means n factorial, the product of all positive integers from 1 to n. For example, 5! = 5×4×3×2×1 = 120. By convention, 0! = 1.

这里 n! 表示 n 的阶乘，即从 1 到 n 的所有正整数之积。例如，5! = 5×4×3×2×1 = 120。按照约定，0! = 1。

Many calculators have an nCr button, but in the exam you should also know how to simplify factorial expressions manually. For instance, 6C2 = 6!/(2!4!) = (6×5)/(2×1) = 15.

许多计算器都有 nCr 按键，但在考试中你还应该掌握如何手动化简阶乘表达式。例如，6C2 = 6!/(2!4!) = (6×5)/(2×1) = 15。

6. Finding a Specific Term | 求特定项

Instead of expanding fully, exam questions often ask for a single term. The general term in the expansion of (a+b)ⁿ is:

考试题目并不总是要求完全展开，而是经常让你求某一特定项。在 (a+b)ⁿ 的展开式中，通项为：

Tₖ₊₁ = nCk aⁿ⁻ᵏ bᵏ

To find the term containing a specific power, set the exponent of the variable equal to the required value and solve for k. Then substitute back into the general term formula.

要求含有特定幂的项，先令变量的指数等于题设要求的值，解出 k。然后将 k 代回通项公式即可。

For example, in the expansion of (x² + 2/x)⁶, the general term is 6Ck (x²)⁶⁻ᵏ (2x⁻¹)ᵏ. The power of x is 12−2k−k = 12−3k. If you need the term in x³, solve 12−3k = 3, giving k = 3.

例如，在 (x² + 2/x)⁶ 的展开式中，通项为 6Ck (x²)⁶⁻ᵏ (2x⁻¹)ᵏ。x 的指数是 12−2k−k = 12−3k。如果要求含 x³ 的项，解 12−3k = 3，得 k = 3。

7. Binomial Expansion with Negative or Fractional Powers | 负指数或分数指数的二项展开

When n is not a positive integer, such as n = −1, 1/2, or −2/3, the expansion becomes an infinite series. The formula is valid for (1+x)ⁿ when |x| < 1, and it is:

当 n 不是正整数时，例如 n = −1、1/2 或 −2/3，展开式成为无穷级数。公式对 (1+x)ⁿ 在 |x| < 1 时成立，形式为：

(1+x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + …

This expansion never terminates. In A-Level questions, you are usually asked to expand up to the term in x² or x³ and state the range of validity.

这个展开式永远不会终止。在 A-Level 题目中，通常要求你展开到 x² 或 x³ 项，并给出有效范围。

For example, expand (1+3x)⁻¹ up to x³: first write it as (1+u)⁻¹ with u = 3x, then use the formula with n = −1.

例如，将 (1+3x)⁻¹ 展开到 x³：先将其写成 (1+u)⁻¹ 的形式，其中 u = 3x，然后用 n = −1 的公式。

(1+3x)⁻¹ = 1 − 3x + 9x² − 27x³ + …

8. Validity Conditions | 有效条件

For the binomial expansion of (1+x)ⁿ with non-integer n, the series is valid only when |x| < 1. If the expression is (a+bx)ⁿ, first rewrite it as aⁿ(1+(b/a)x)ⁿ, and the validity condition becomes |(b/a)x| < 1, that is |x| < |a/b|.

对于非整数 n 的 (1+x)ⁿ 二项展开，级数仅在 |x| < 1 时有效。如果表达式是 (a+bx)ⁿ，先将其改写为 aⁿ(1+(b/a)x)ⁿ，有效条件就变为 |(b/a)x| < 1，即 |x| < |a/b|。

Always state the validity condition in your final answer. A common exam mistake is to expand correctly but forget to give the range of x for which the expansion is valid.

在最终答案中一定要写出有效条件。考试中常见的错误是展开式正确，但忘记给出展开式成立时 x 的范围。

For example, for (4+5x)¹ᐟ², first write it as 2(1+5x/4)¹ᐟ². The expansion is valid when |5x/4| < 1, so |x| < 4/5.

例如，对于 (4+5x)¹ᐟ²，先写成 2(1+5x/4)¹ᐟ²。展开式在 |5x/4| < 1 时有效，即 |x| < 4/5。

9. Approximations using Binomial Expansion | 二项展开式的近似计算

Binomial expansion is often used to approximate values. If a question asks you to estimate √1.02, you can write it as (1+0.02)¹ᐟ² and use the first few terms of the expansion.

二项展开式常被用于近似计算。如果题目让你估算 √1.02，你可以将其写成 (1+0.02)¹ᐟ²，然后使用展开式的前几项。

(1+x)¹ᐟ² ≈ 1 + (1/2)x − (1/8)x² + (1/16)x³

Substituting x = 0.02 gives 1 + 0.01 − 0.00005 + 0.00000025 ≈ 1.00995025. The more terms you keep, the more accurate the approximation.

代入 x = 0.02 得到 1 + 0.01 − 0.00005 + 0.00000025 ≈ 1.00995025。保留的项越多，近似值越精确。

In the exam, always show the substitution step clearly and state whether the approximation is an overestimate or underestimate if required.

在考试中，务必清晰地展示代入步骤，如果题目要求，还要说明该近似是高估还是低估。

10. Exam-style Worked Example | 考试型例题讲解

Consider this typical Edexcel question: Expand (1−2x)⁻¹² up to the term in x³ and state the range of values of x for which the expansion is valid.

考虑一道典型的 Edexcel 题目：将 (1−2x)⁻¹² 展开到 x³ 项，并写出展开式有效的 x 取值范围。

Using the formula with n = −1/2 and u = −2x, we get:

使用 n = −1/2 且 u = −2x 的公式，我们得到：

(1−2x)⁻¹² = 1 + (−1/2)(−2x) + [(−1/2)(−3/2)/2!](−2x)² + [(−1/2)(−3/2)(−5/2)/3!](−2x)³ + …

Simplify each term: first term 1, second term +x, third term + (3/8)(4x²) = (3/2)x², fourth term + (−15/48)(−8x³) = (5/2)x³? Let me check carefully: third term coefficient = (−1/2)(−3/2)/2 = 3/4, times (−2x)² = 4x² gives 3x². Wait, let’s recalculate properly.

化简每一项：第一项 1，第二项 +x。第三项的系数为 (−1/2)(−3/2)/2 = 3/8? Actually (−1/2)(−3/2) = 3/4, divided by 2! = 2 gives 3/8. Then times (−2x)² = 4x² gives (3/8)(4x²) = (3/2)x². 第四项：系数 (−1/2)(−3/2)(−5/2)/3! = (−15/8)/6 = −15/48 = −5/16. 乘以 (−2x)³ = −8x³ 得 (−5/16)(−8x³) = (5/2)x³. So the expansion is:

(1−2x)⁻¹² = 1 + x + (3/2)x² + (5/2)x³ + …

The validity condition is |−2x| < 1, so |x| < 1/2. Always write this final line.

有效条件为 |−2x| < 1，即 |x| < 1/2。务必写出这最后一行。

11. Common Mistakes to Avoid | 常见错误提醒

Forgetting to adjust the validity condition when the coefficient of x is not 1.
Mixing up nCr with nPr or using the wrong factorial denominator.
Making sign errors when the binomial contains a negative term such as (1−x)ⁿ.
Expanding only the first two terms and forgetting the later required terms.
Using the expansion for non-integer n outside its range of validity.

当 x 的系数不是 1 时，忘记调整有效条件。
混淆 nCr 和 nPr，或使用错误的分母阶乘。
二项式中含有负项（如 (1−x)ⁿ）时出现符号错误。
只展开前两项，忘记题目要求的后续项。
在非整数 n 的展开式中，超出有效范围使用公式。

Reading the question carefully and writing down the general term first will help you avoid most of these errors. Practise past paper questions under timed conditions to build confidence.

仔细审题并先写出通项，有助于避免上述大部分错误。在限时条件下练习往年真题，可以增强应试信心。

12. Key Takeaways | 要点总结

Binomial expansion is a high-yield topic in Edexcel A-Level Maths. Remember the general term nCk aⁿ⁻ᵏ bᵏ, know how to find a specific term, and always state the validity condition for non-integer expansions.

二项展开式是 Edexcel A-Level 数学中分值较高的考点。记住通项 nCk aⁿ⁻ᵏ bᵏ，学会求特定项，并且对于非整数展开务必写出有效条件。

Mastering this topic will also help with sequences and series, approximations, and later calculus topics. Review the worked example and try similar questions from past Edexcel papers.

掌握这一主题也有助于学习数列与级数、近似计算以及后续微积分内容。复习例题并尝试 Edexcel 历年试卷中的类似题目。

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Binomial Expansion for A-Level Edexcel Maths | A-Level Edexcel 数学二项展开式精讲