📚 Exercise 24E.2 | 练习24E.2
Exercise 24E.2 focuses on applying the binomial theorem to find specific terms, constant terms, and coefficients in the expansion of binomial expressions with positive integer powers. Mastering this exercise is essential for IB Mathematics examinations, as it tests not only algebraic manipulation but also a deep understanding of combinatorial coefficients. Students will learn to use the general term formula efficiently and avoid common pitfalls when handling negative signs, fractions within the binomial, or multiple products.
练习 24E.2 集中于运用二项式定理，求正整数幂的二项展开式中的特定项、常数项及系数。掌握这部分内容对 IB 数学考试至关重要，因为它不仅考查代数运算能力，还检验对组合系数的深刻理解。学生将学会高效利用通项公式，并避免在处理负号、二项式内部分式或多个乘积时常见的错误。
1. Understanding the Binomial Theorem | 理解二项式定理
For a positive integer n, the binomial theorem states that (a + b)ⁿ = Σⁿᵣ₌₀ C(n, r) aⁿ⁻ʳ bʳ, where C(n, r) = n! / [r!(n−r)!] is the binomial coefficient.
对于正整数 n，二项式定理表述为 (a + b)ⁿ = Σⁿᵣ₌₀ C(n, r) aⁿ⁻ʳ bʳ，其中 C(n, r) = n! / [r!(n−r)!] 是二项式系数。
Each term in the expansion is formed by taking a and b with exponents that add up to n, multiplied by the corresponding binomial coefficient. The number of terms is n+1.
展开式中的每一项由 a 和 b 的指数之和为 n 构成，并乘以相应的二项式系数。总项数为 n+1。
2. The General Term Formula | 通项公式
The (r+1)th term of the expansion is given by Tr+1 = C(n, r) an−r br, where r takes integer values from 0 to n. This formula allows us to target a specific term without expanding the entire expression.
展开式的第 r+1 项由公式 Tr+1 = C(n, r) an−r br 给出，其中 r 取 0 到 n 的整数。该公式使我们无需展开整个表达式即可锁定特定项。
In Exercise 24E.2, you will often need to set the exponent of the variable to zero (for constant terms) or to a desired power, then solve for r.
在练习 24E.2 中，经常需要令变量的指数为零（求常数项）或等于所需的幂次，然后解出 r。
3. Finding a Specific Term – Step by Step | 求特定项 – 步骤解析
First, write the general term Tr+1 clearly, simplifying powers and collecting coefficients. Next, impose the condition on the exponent of the variable you are interested in. Solve the resulting equation to find the valid integer r. Finally, substitute r back into the general term to compute the coefficient or the entire term.
首先，清晰地写出通项 Tr+1，化简幂次并合并系数。然后，对你感兴趣的变量的指数施加条件。求解所得的方程，找到有效的整数 r。最后，将 r 代回通项，计算出系数或整个项。
If the equation for r does not yield an integer between 0 and n, then the specified term does not exist in the expansion. This logic check is crucial in avoiding unnecessary work.
如果关于 r 的方程未得到 0 到 n 之间的整数解，说明指定项在该展开式中不存在。这一逻辑检验对于避免无用功至关重要。
4. Example 1: Constant Term in (x + 2/x)⁸ | 示例1：(x + 2/x)⁸ 中的常数项
We rewrite the binomial as (x + 2x⁻¹)⁸. The general term is Tr+1 = C(8, r) x8−r (2x⁻¹)r = C(8, r) 2ʳ x8−2r.
我们将二项式改写为 (x + 2x⁻¹)⁸。通项为 Tr+1 = C(8, r) x8−r (2x⁻¹)r = C(8, r) 2ʳ x8−2r。
ust be zero, so 8 − 2r = 0 → r = 4. The constant term is T₅ = C(8, 4) 2⁴ = 70 × 16 = 1120.
要求常数项，则 x 的指数必须为零，故 8 − 2r = 0 → r = 4。常数项为 T₅ = C(8, 4) 2⁴ = 70 × 16 = 1120。
Tr+1 = C(8, r) 2ʳ x8−2r → constant term 1120
5. Example 2: Coefficient of x³ in (2x − 3)⁷ | 示例2：(2x − 3)⁷ 中 x³ 的系数
Here a = 2x, b = −3. The general term is Tr+1 = C(7, r) (2x)7−r (−3)r = C(7, r) 27−r (−3)ʳ x7−r.
这里 a = 2x, b = −3。通项为 Tr+1 = C(7, r) (2x)7−r (−3)r = C(7, r) 27−r (−3)ʳ x7−r。
so 7 − r = 3 → r = 4. The required coefficient is C(7, 4) 2³ (−3)⁴ = 35 × 8 × 81 = 22680.
我们需要 x 的指数为 3，故 7 − r = 3 → r = 4。所求系数为 C(7, 4) 2³ (−3)⁴ = 35 × 8 × 81 = 22680。
Notice that the negative base is raised to an even power, making the sign positive. Had r been odd, the sign would be negative.
注意负底数的幂次为偶数，符号为正。若 r 为奇数，则符号为负。
6. Working with Coefficients and Multiple Variables | 处理系数与多变量
When the binomial contains coefficients, such as (axᵖ + byᵠ)ⁿ, the general term becomes Tr+1 = C(n, r) (axᵖ)n−r (byᵠ)r = C(n, r) an−r bʳ xp(n−r) yq r. You can then set conditions on the exponents of x and y simultaneously.
当二项式中包含系数，如 (axᵖ + byᵠ)ⁿ，通项变为 Tr+1 = C(n, r) (axᵖ)n−r (byᵠ)r = C(n, r) an−r bʳ xp(n−r) yq r。然后可以同时对 x 和 y 的指数施加条件。
nt of x to zero: 2(9−r) + (−1)r = 0 → 18 − 3r = 0 → r = 6. This method is highly reliable.
例如，求 (3x² + 1/x)⁹ 中与 x 无关的项，令 x 的总指数为零：2(9−r) + (−1)r = 0 → 18 − 3r = 0 → r = 6。这一方法非常可靠。
7. Sum of Coefficients and Special Values | 系数和与特殊值
ng a = 1 and b = 1, i.e., (1+1)ⁿ = 2ⁿ, can provide quick checks or solve related problems.
虽然练习 24E.2 主要针对单个项，但了解展开式所有系数之和可通过令 a = 1 与 b = 1 得到，即 (1+1)ⁿ = 2ⁿ，这能提供快速检验或解决相关问题。
yields the alternating sum of coefficients, useful for isolating certain subsets of terms.
类似地，令 a = 1, b = −1 可得到系数的交错和，有助于分离特定的项子集。
8. Common Mistakes in Exercise 24E.2 | 练习24E.2中常见错误
One frequent error is forgetting to apply the exponent to the entire factor b, especially when b is a product like 2/x. Students sometimes write (2/x)ʳ as 2 x⁻ʳ instead of 2ʳ x⁻ʳ. Always distribute the outer exponent first.
一个常见错误是忘记将指数应用于整个因式 b，特别是当 b 为 2/x 这样的乘积时。学生有时会把 (2/x)ʳ 写成 2 x⁻ʳ 而不是 2ʳ x⁻ʳ。务必先将外部指数分配到所有因子上。
Another mistake is misidentifying the index r in the binomial coefficient. Tr+1 e term numbering starts at r = 0 for the first term. Keep a systematic record of r values.
另一个错误是二项式系数中的索引 r 识别有误。Tr+1 使用的是 C(n, r)，而非 C(n, r+1)。项编号从 r = 0 开始对应第一项。请系统记录 r 值。
9. Extension: Product of Two Binomial Expansions | 拓展：两个二项展开式的乘积
Sometimes problems require finding a specific term in the product of two binomial expansions, e.g., (1+x)⁵ (1+2x)⁶. The strategy is to expand each partially to the required degree, then multiply only the combinations that yield the target power.
有时题目要求求出两个二项展开式乘积中的特定项，例如 (1+x)⁵ (1+2x)⁶。策略是各自部分展开至所需次数，然后仅将能产生目标幂次的组合相乘。
This approach is far more efficient than full expansion and is a typical extension in Exercise 24E.2-type assignments. Use a table or systematic pairing to ensure you capture all contributions.
这种方法比完全展开高效得多，是练习 24E.2 类作业的典型拓展。可使用表格或系统配对以确保捕获所有贡献项。
10. Practice Tips and Summary | 练习建议与总结
sion first, then solve for r. Check that the solution for r is a non-negative integer ≤ n.
开始时明确 a、b 和 n。代入数字之前先写出通项。始终先简化指数表达式，再解 r。检验 r 的解是否为不大于 n 的非负整数。
Regular practice with varied forms, such as (kx + 1/x²)ⁿ or (√x − 1/√x)ⁿ, develops fluency. Use Exercise 24E.2 as a diagnostic tool: if you can confidently find constant terms, greatest coefficients, and two specific terms in mixed questions, you are ready for IB exams.
通过多变的形式进行定期练习，如 (kx + 1/x²)ⁿ 或 (√x − 1/√x)ⁿ，可培养熟练度。将练习 24E.2 作为诊断工具：如果你能自信地在混合题中找出常数项、最大系数以及两个特定项，就说明已为 IB 考试做好准备。
The binomial theorem is a powerful compact tool: learn to trust the algebra and avoid excessive numerical expansion. Precision in indexing and exponents is key.
二项式定理是一个强大而紧凑的工具：学会信任代数过程，避免过度的数值展开。指数和索引的精确性是关键。
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