Solving Binomial Problems | 二项式问题解题全攻略

📚 Solving Binomial Problems | 二项式问题解题全攻略

Binomial expansions are a key topic in the Edexcel A-Level Mathematics syllabus. Mastering them unlocks many problems in algebra, series, and approximation. This guide provides a step-by-step approach to solving various types of binomial problems you will meet in exams.

二项式展开是 Edexcel A-Level 数学大纲中的关键主题。掌握它能够帮助你解决代数、级数和近似计算中的许多问题。本指南将提供一套循序渐进的方法，用来处理考试中遇到的各种二项式问题。

1. Binomial Expansion for Positive Integer n | 正整数n的二项式展开

For any positive integer n, the binomial theorem gives the expansion of (a + b)ⁿ as a sum of terms involving powers of a and b. The standard formula is:

对任意正整数 n，二项式定理将 (a + b)ⁿ 展开为若干项之和，每一项包含 a 和 b 的幂。标准公式为：

(a + b)ⁿ = ∑_r=0ⁿ C(n,r) a^n−r b^r

Here C(n,r) is the binomial coefficient, also written as nCr or “n choose r”. The expansion has n+1 terms.

这里的 C(n,r) 称为二项式系数，也可写作 nCr 或 “n 选 r”。展开式共有 n+1 项。

For example, (a + b)³ = a³ + 3a²b + 3ab² + b³. The coefficients 1, 3, 3, 1 follow the pattern from Pascal’s triangle.

例如，(a + b)³ = a³ + 3a²b + 3ab² + b³。系数 1, 3, 3, 1 遵循帕斯卡三角的规律。

2. Understanding Factorials and Combinations | 理解阶乘与组合

The binomial coefficient C(n,r) is defined using factorials. The factorial of a positive integer n, written n!, is the product of all positive integers less than or equal to n.

二项式系数 C(n,r) 通过阶乘定义。正整数 n 的阶乘写作 n!，它是所有不大于 n 的正整数的乘积。

n! = n × (n−1) × (n−2) × … × 2 × 1

The combination formula is:

组合公式为：

C(n,r) = n! / (r! (n−r)!)

For example, C(6,2) = 6! / (2! × 4!) = (6 × 5) / (2 × 1) = 15. Always simplify the factorial expression before doing heavy multiplication.

例如，C(6,2) = 6! / (2! × 4!) = (6 × 5) / (2 × 1) = 15。在大量乘法之前，一定要先化简阶乘表达式。

Note that C(n,0) = C(n,n) = 1 and C(n,1) = n. These special values appear often in binomial problems.

注意 C(n,0) = C(n,n) = 1，且 C(n,1) = n。这些特殊值在二项式问题中经常出现。

3. Finding a Specific Term | 求特定项

In the expansion of (a + b)ⁿ, the term containing b^r is the (r+1)th term. It is given by:

在 (a + b)ⁿ 的展开式中，含有 b^r 的项是第 (r+1) 项，其表达式为：

T_r+1 = C(n,r) a^n−r b^r

To find a particular term, identify the required power r and then substitute into the formula.

要求某一特定项，先确定所需的幂 r，然后代入公式即可。

Example: Find the 4th term in the expansion of (x + 2)⁶. Here n=6, r=3 (since the 4th term has r=3).

例：求 (x + 2)⁶ 展开式中的第 4 项。这里 n=6，r=3（因为第 4 项对应 r=3）。

T₄ = C(6,3) x⁶⁻³ (2)³ = 20 × x³ × 8 = 160x³

Always check whether the term is “the 4th term” (r=3) or “the term in x³” (also r=3).

始终要确认是“第 4 项”（r=3），还是“x³ 的项”（也是 r=3）。

4. Finding the Coefficient of x^k | 求 x^k 的系数

When the expression is of the form (ax + b)ⁿ, each term is C(n,r) (ax)^n−r b^r = C(n,r) a^n−r b^r x^n−r. To find the coefficient of x^k, set the power of x equal to k.

当式子形如 (ax + b)ⁿ 时，每一项为 C(n,r) (ax)^n−r b^r = C(n,r) a^n−r b^r x^n−r。要求 x^k 的系数，只需令 x 的指数等于 k。

Example: Find the coefficient of x³ in (1 + 2x)⁵.

例：求 (1 + 2x)⁵ 中 x³ 的系数。

Here a=1, b=2x, n=5. The general term is C(5,r) (2x)^r = C(5,r) 2^r x^r. Set r=3 to get x³.

这里 a=1, b=2x, n=5。通项为 C(5,r) (2x)^r = C(5,r) 2^r x^r。令 r=3 得到 x³ 的项。

Coefficient = C(5,3) × 2³ = 10 × 8 = 80

If the bracket has two x-terms, such as (2 − 3x)⁴, the general term is C(4,r) (2)^4−r (−3x)^r. The x power is r, so the x² coefficient uses r=2.

如果括号内有两个含 x 的项，例如 (2 − 3x)⁴，通项为 C(4,r) (2)^4−r (−3x)^r。x 的幂为 r，因此求 x² 的系数时令 r=2。

C(4,2) × 2² × (−3)² = 6 × 4 × 9 = 216

5. Binomial Expansion with Unknown n | 含未知数n的展开

Sometimes the power n is unknown but the coefficient of a given term is provided. You can set up an equation and solve for n.

有时幂 n 未知，但给出了某一项的系数。这时可以建立方程并求解 n。

Example: In the expansion of (1 + x)ⁿ, the coefficient of x² is 28. Find n, where n is a positive integer.

例：在 (1 + x)ⁿ 的展开式中，x² 的系数为 28。求正整数 n。

The coefficient of x² is C(n,2) = n(n−1)/2. So n(n−1)/2 = 28, giving n(n−1) = 56.

x² 的系数为 C(n,2) = n(n−1)/2。因此 n(n−1)/2 = 28，即 n(n−1) = 56。

n² − n − 56 = 0 ⇒ (n−8)(n+7) = 0 ⇒ n = 8 (since n > 0)

Always reject negative or non-integer solutions if n is stated to be a positive integer.

如果题目说明 n 为正整数，一定要舍去负数或非整数解。

6. Using Binomial Expansions for Approximations | 用二项式展开近似计算

For small values of x, the first few terms of a binomial expansion give a good approximation. This is especially useful when calculating powers of numbers close to 1.

当 x 很小时，二项式展开的前几项就能给出很好的近似值。这在计算接近 1 的数的幂时特别有用。

For (1 + x)ⁿ, the expansion begins as:

对于 (1 + x)ⁿ，展开式开头为：

(1 + x)ⁿ = 1 + n x + n(n−1)/2 x² + n(n−1)(n−2)/6 x³ + …

Example: Approximate (1.01)¹⁰ by setting x=0.01 and n=10. Use the first three terms.

例：令 x=0.01，n=10，用前三项近似计算 (1.01)¹⁰。

(1.01)¹⁰ ≈ 1 + 10(0.01) + 45(0.01)² = 1 + 0.1 + 0.0045 = 1.1045

The actual value is about 1.1046, so the approximation is accurate to 3 decimal places. More terms give even better accuracy.

实际值约为 1.1046，因此该近似精确到小数点后 3 位。项数越多，近似越精确。

7. Validity of Expansion for Rational Exponents | 有理数指数展开的有效性

When n is not a positive integer (e.g. n = −1 or n = ½), the binomial expansion becomes an infinite series. It is then valid only when |x| < 1 (for (1 + x)ⁿ). These expansions are often used to find series representations of rational and radical functions.

当 n 不是正整数（如 n = −1 或 n = ½）时，二项式展开变成无穷级数。此时对于 (1 + x)ⁿ，仅当 |x| < 1 时级数收敛。这些展开常用于求有理函数和根式函数的级数表示。

For example:

例如：

(1 + x)⁻¹ = 1 − x + x² − x³ + … , |x| < 1

(1 + x)^(½) = 1 + ½ x − ⅛ x² + … , |x| < 1

When substituting a value of x, always check that |x| is within the radius of convergence. For example, to approximate √0.98, write √(1−0.02), so x = −0.02 which satisfies |x| < 1.

代入 x 值时，务必检查 |x| 是否在收敛半径内。例如，要近似 √0.98，可写成 √(1−0.02)，于是 x = −0.02，满足 |x| < 1。

8. Problem-Solving Strategies and Common Mistakes | 解题策略与常见错误

Here are some strategies to avoid common pitfalls in binomial problems.

以下是一些避免常见错误的解题策略。

Check the number of terms: for positive integer n, there are n+1 terms. The first term has r=0 and the last has r=n.

检查项数：对于正整数 n，展开式有 n+1 项。第一项对应 r=0，最后一项对应 r=n。
Be careful with negative signs: if the second term is negative, the sign alternates. For (a − b)ⁿ, the terms are (+ − + − …).

注意负号：如果第二项是负的，符号会交替变化。对于 (a − b)ⁿ，各项符号依次为 (+ − + − …)。
Do not confuse permutation P(n,r) with combination C(n,r). In binomial expansion, order does not matter, so use combinations.

不要混淆排列 P(n,r) 与组合 C(n,r)。在二项式展开中，顺序不重要，因此使用组合。
When finding a coefficient, remember to include the coefficients from the bracket. For (2 + 3x)⁵, the coefficient of x² includes 2³ × 3², not just C(5,2).

求系数时，不要忘记括号内各项的系数。例如 (2 + 3x)⁵ 中 x² 的系数包含 2³ × 3²，而不仅仅是 C(5,2)。
For rational exponents, always state the range of validity |x| < 1 if the question asks for it.

对于有理数指数，如果题目要求，一定要指出有效范围 |x| < 1。

9. Worked Examples | 典型例题精讲

Let us apply these methods to three typical exam-style problems.

下面通过三个典型的考试风格例题来应用这些方法。

Example 1: Find the constant term in the expansion of (x + 2/x²)⁶.

例 1：求 (x + 2/x²)⁶ 展开式中的常数项。

The general term is C(6,r) x^6−r (2/x²)^r = C(6,r) 2^r x^6−r−2r = C(6,r) 2^r x^6−3r.

通项为 C(6,r) x^6−r (2/x²)^r = C(6,r) 2^r x^6−r−2r = C(6,r) 2^r x^6−3r。

For the constant term, the exponent of x must be 0, so 6−3r = 0 ⇒ r = 2.

常数项要求 x 的指数为 0，所以 6−3r = 0 ⇒ r = 2。

Constant term = C(6,2) 2² = 15 × 4 = 60

Example 2: In the expansion of (1 + ax)ⁿ, the coefficient of x is 12 and the coefficient of x² is 60. Find n and a.

例 2：在 (1 + ax)ⁿ 的展开式中，x 的系数为 12，x² 的系数为 60。求 n 和 a。

The x coefficient is n a = 12. The x² coefficient is C(n,2) a² = n(n−1)a²/2 = 60.

x 的系数为 n a = 12。x² 的系数为 C(n,2) a² = n(n−1)a²/2 = 60。

From na=12, a=12/n. Substitute into the second equation:

由 na=12，得 a=12/n。代入第二个方程：

n(n−1)/2 × (144/n²) = 60 ⇒ 72(n−1)/n = 60 ⇒ 72n − 72 = 60n ⇒ 12n = 72 ⇒ n = 6

Then a = 12/6 = 2. So n=6 and a=2.

于是 a = 12/6 = 2。因此 n=6，a=2。

Example 3: Use the binomial expansion to find (0.99)⁵ correct to 4 decimal places.

例 3：利用二项式展开求 (0.99)⁵ 精确到小数点后 4 位。

Write 0.99 = 1 − 0.01, so (0.99)⁵ = (1 − 0.01)⁵. Expand up to the cube term:

将 0.99 写成 1 − 0.01，即 (0.99)⁵ = (1 − 0.01)⁵。展开到立方项：

(1 − x)⁵ = 1 − 5x + 10x² − 10x³ + …

With x=0.01:

代入 x=0.01：

≈ 1 − 0.05 + 10(0.0001) − 10(0.000001) = 1 − 0.05 + 0.001 − 0.00001 = 0.95099

Rounded to 4 decimal places, (0.99)⁵ ≈ 0.9510.

四舍五入到小数点后 4 位，(0.99)⁵ ≈ 0.9510。

10. Practice Questions | 练习

Test your understanding with these questions. Answers are provided below.

用以下问题测试你的理解。答案附在后面。

1. Find the coefficient of x⁴ in (2 + x)⁷.

1. 求 (2 + x)⁷ 中 x⁴ 的系数。
2. In the expansion of (1 + kx)¹⁰, the coefficient of x² is 315. Find the positive value of k.

2. 在 (1 + kx)¹⁰ 的展开式中，x² 的系数为 315。求 k 的正值。
3. Determine the exact value of the constant term in (3x² − 1/(2x))⁹.

3. 求 (3x² − 1/(2x))⁹ 展开式中常数项的确切值。
4. Use the binomial series for (1 + x)^(−2) to write down the first three non-zero terms and state the range of validity.

4. 利用 (1 + x)^(−2) 的二项式级数写出前三个非零项，并指出有效范围。

Answers: 1. 560; 2. k=1.5; 3. 189/16; 4. 1 − 2x + 3x², |x| < 1.

答案：1. 560；2. k=1.5；3. 189/16；4. 1 − 2x + 3x²，|x| < 1。

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Solving Binomial Problems | 二项式问题解题全攻略