📚 Mixed Exercise 8: Binomial Expansion | 混合练习8：二项式展开
Mixed Exercise 8 in the Edexcel AS Pure Mathematics textbook reinforces one of the most versatile tools in algebra – the binomial expansion for positive integer powers. This mixed exercise challenges students to identify coefficients, work with unknown constants, and use expansions to estimate values efficiently. A clear command of the binomial theorem not only secures marks on straightforward questions but also underpins later topics in calculus and series. In this article, we will revisit the core principles, dissect typical exam-style problems, and offer strategies to avoid common pitfalls.
Edexcel AS 纯数学教材中的混合练习8 巩固了代数中用途最广泛的工具之一——正整数幂的二项式展开。这份混合练习要求学生找系数、处理未知常数，并利用展开式高效地估算数值。透彻掌握二项式定理不仅能确保简单题目的得分，也为后续的微积分与级数内容打下基础。在本文中，我们将重温核心原理，剖析典型的考试题型，并提供避免常见错误的策略。
1. The Binomial Theorem for Positive Integers | 正整数情况下二项式定理
itive integer n, the expansion of (a + b)ⁿ is given by (a + b)ⁿ = Σ (r=0 to n) C(n, r) aⁿ⁻ʳ bʳ, where C(n, r) = n! / [r! (n - r)! , the second is C(n, 1) aⁿ⁻¹ b, and the pattern continues until the final term C(n, n) bⁿ.
对于任意正整数 n，(a + b)ⁿ 的展开式为 (a + b)ⁿ = Σ (r=0 到 n) C(n, r) aⁿ⁻ʳ bʳ，其中 C(n, r) = n! / [r! (n - r)!] 是二项式系数。求和从 r = 0 进行到 n，共产生 n + 1 项。首项为 C(n, 0) aⁿ，第二项为 C(n, 1) aⁿ⁻¹ b，以此类推，末项为 C(n, n) bⁿ。
When the binomial takes the form (1 + x)ⁿ, the coefficients simplify to C(n, r), and the expansion becomes 1 + nx + [n(n - 1)/2!] x² + ... This structure is extremely common in Mixed Exercise 8 because it allows easy manipulation of unknown constants and estimation of decimal powers.
当二项式为 (1 + x)ⁿ 时，系数简化为 C(n, r)，展开式变为 1 + nx + [n(n - 1)/2!] x² + ... 这种结构在混合练习8中极为常见，因为它便于处理未知常数及估算小数次幂。
2. Pascal's Triangle and Combinatorial Coefficients | 帕斯卡三角形与组合系数
Binomial coefficients can be quickly obtained from Pascal' two above it, mirroring the identity C(n, r) = C(n - 1, r - 1) + C(n - 1, r). While Pascal's triangle is convenient for n ≤ 8, using the factorial formula C(n, r) = n! / [r! (n - r)!] is essential for larger n or when determining coefficients algebraically.
对于较小的 n 值，二项式系数可以通过帕斯卡三角形快速获取。每个数等于其上方两数之和，这反映了恒等式 C(n, r) = C(n - 1, r - 1) + C(n - 1, r)。帕斯卡三角形在 n ≤ 8 时很方便，但当 n 较大或需要利用代数方法求系数时，就必须使用阶乘公式 C(n, r) = n! / [r! (n - r)!]。
s is where the general term formula is most useful, rather than drawing out rows of Pascal's triangle. Always remember that C(n, r) = C(n, n - r), so coefficients are symmetric.
在混合练习8中，你常常被要求找到某一特定项而无需完全展开。这时通项公式最为有用，而非逐行画出帕斯卡三角形。始终牢记 C(n, r) = C(n, n - r)，因此系数是对称分布的。
3. The General Term and Its Components | 通项及其各部分
The (r + 1)-th term in the expansion of (a + b)ⁿ is Tr+1 = C(n, r) aⁿ⁻ʳ bʳ e constant term. For an expression like (2 - 3x)⁵, the term containing x³ corresponds to r = 3, giving T₄ = C(5, 3) · 2² · (-3x)³.
(a + b)ⁿ 展开式的第 r + 1 项为 Tr+1 = C(n, r) aⁿ⁻ʳ bʳ。其中 r 为索引，首项的 r = 0。当问题要求求含 x⁴ 的项或常数项时，正确分辨 r 的值至关重要。例如在 (2 - 3x)⁵ 中，含 x³ 的项对应 r = 3，即 T₄ = C(5, 3) · 2² · (-3x)³。
Many mistakes occur because students use r equal to the desired power of x. For (1 + 2x)ⁿ, the term in xᵏ is C(n, k) (2x)ᵏ. In this case, the power of x directly matches r, but careful attention is needed when the binomial contains a factor like (ax + b)ⁿ or when the variable appears in both a and b.
许多错误源于学生令 r 直接等于所求 x 的幂次。对 (1 + 2x)ⁿ，含 xᵏ 的项是 C(n, k) (2x)ᵏ。此时 x 的幂次恰好对应 r，但当二项式中含有如 (ax + b)ⁿ 或因变量同时出现在 a 和 b 中时，需要格外小心。
4. Finding a Specific Coefficient | 求特定系数
To find the coefficient of xᵐ in (p + qx)ⁿ, set the term Tr+1 m. Then evaluate C(n, m) · pⁿ⁻ᵐ · qᵐ. If the expression is something like (1 + ax)(1 + bx)ⁿ, you may need to consider contributions from two different terms. Mixed Exercise 8 regularly includes such multi-stage coefficient problems.
求 (p + qx)ⁿ 中 xᵐ 的系数时，列出通项 Tr+1 并令 x 的幂次等于 m：解得 r = m。然后计算 C(n, m) · pⁿ⁻ᵐ · qᵐ。若表达式形如 (1 + ax)(1 + bx)ⁿ，你可能需要考虑来自不同项的贡献。混合练习8经常包含这类多步骤的系数题。
For example, to find the coefficient of x² in (1 + 3x)(2 - x)⁴, expand (2 - x)⁴ up to x², multiply by (1 + 3x), and collect like terms. The ability to combine expansions efficiently is tested to distinguish high-performing students.
例如，求 (1 + 3x)(2 - x)⁴ 中 x² 的系数，可先将 (2 - x)⁴ 展开至 x² 项，再乘以 (1 + 3x) 并合并同类项。能否高效合并展开式正是区分优秀学生的考查点。
5. Expansions with Unknown Constants | 含未知常数的展开
aring coefficients gives nC1 · a = 12 and nC2 · a² = 64. This yields the system na = 12 and [n(n - 1)/2] a² = 64. Substituting a = 12/n into the second equation leads to a quadratic in n that can be solved to find integer values.
混合练习8中有一类经典问题，会给出 (1 + ax)ⁿ 的前三项并要求解出 a 和 n 的值。假设 (1 + ax)ⁿ = 1 + 12x + 64x² + ...，比较系数得到 nC1 · a = 12 以及 nC2 · a² = 64。由此建立方程组 na = 12 与 [n(n - 1)/2] a² = 64。将 a = 12/n 代入第二个方程，可化为关于 n 的二次方程，解出整数值。
ivalent valid solution). This technique relies on understanding that the coefficients of x and x² come from the general term. In the exam, always verify that your values produce the given sequence and check for extraneous solutions, as n must be a positive integer.
解得 n = 9，a = 4/3（或其他合理解）。这一方法依赖于理解 x 和 x² 的系数来自通项。考试中务必验证所求值能否产生题给序列，并排除增解，因为 n 必须是正整数。
6. Expanding (a + bx)ⁿ When a ≠ 1 | 当 a ≠ 1 时 (a + bx)ⁿ 的展开
)ⁿ as aⁿ(1 + (b/a)x)ⁿ, which is correct, but then misapply the general term. The term in xʳ is C(n, r) aⁿ⁻ʳ (bx)ʳ. When a is not 1, the coefficient of xʳ involves aⁿ⁻ʳ value. In Mixed Exercise 8, questions often set a = 2 or a = 3 to test this understanding.
许多学生误将 (a + bx)ⁿ 写成 aⁿ(1 + (b/a)x)ⁿ，这本身正确，但随后却错误地应用通项。xʳ 项的表达式为 C(n, r) aⁿ⁻ʳ (bx)ʳ。当 a 不为 1 时，xʳ 的系数包含 aⁿ⁻ʳ，这会极大地影响数值。混合练习8中的题目常设 a = 2 或 a = 3 来考查这一理解。
080x³. Notice how the powers of 2 and 3 combine. Overlooking the factor 2² is a common error. Always write the full term and evaluate systematically.
例如，在 (2 + 3x)⁵ 中，含 x³ 的项为 C(5, 3) · 2² · (3x)³ = 10 · 4 · 27x³ = 1080x³。注意 2 的幂次与 3 的幂次如何结合。忽略 2² 因子是一个常见错误。请务必写出完整的项，并系统地求值。
7. Using Binomial Expansions for Approximations | 利用二项式展开进行近似
1 + x)¹⁰ ≈ 1 + 10x + 45x², and substituting gives 1 + 0.02 + 0.00018 = 1.02018, which is accurate to several decimal places. Mixed Exercise 8 includes approximation problems that link to the idea of neglecting higher powers.
二项式展开可以用来估算 (1.002)¹⁰ 或 (0.99)⁸ 等值，方法是将其写成 (1 + x)ⁿ 且 x 很小。对于 (1.002)¹⁰，令 x = 0.002；则 (1 + x)¹⁰ ≈ 1 + 10x + 45x²，代入得 1 + 0.02 + 0.00018 = 1.02018，精确到小数点后若干位。混合练习8包含近似计算问题，体现了忽略高次幂的思路。
When approximating, you must state why higher‑order terms are omitted – usually because x is small enough that x³ and above are negligible. The question may specify the degree of accuracy or ask you to find the percentage error. Always use the exact expansion up to the required power and show the substitution clearly.
在近似时，必须说明为何省略高阶项——通常是因为 x 足够小，x³ 及以上项可忽略不计。题目可能会指定精确度或要求计算百分误差。始终使用展开到所需次幂的精确形式，并清晰地展示代入过程。
8. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One frequent error is misidentifying the value of r when the term order is requested. Remember: the term number is r + 1, not r. Another mistake is forgetting to raise a constant coefficient inside the binomial to the correct power. For (3 + x)⁶, the term in x² is C(6, 2) · 3⁴ · x², not C(6, 2) · 3⁶⁻²· x², but students often use 3⁶⁻² incorrectly or forget to apply the power to 3 entirely.
一个常见错误是当要求项的顺序时，错误地确定 r 值。记住：项数为 r + 1，而非 r。另一个错误是忘记将二项式内的常数系数取正确的次幂。对于 (3 + x)⁶，x² 项为 C(6, 2) · 3⁴ · x²，而非 C(6, 2) · 3⁶⁻² · x²，但学生经常错误地使用 3⁶⁻² 或完全忘了对 3 取幂。
t could be zero without checking. In an equation like n(n - 1)a² = 128, keep factors together and look for integer solutions. Finally, always simplify coefficients fully – leaving 10 · 4 · 27 as an answer loses marks in Edexcel papers.
使用表格系统地列出各次幂可以防止算术差错。此外，求解未知常数时，不要不经检查就除以可能为零的变量。在形如 n(n - 1)a² = 128 的方程中，应保留因式并寻找整数解。最后，务必将系数充分化简——把 10 · 4 · 27 留作最终答案在 Edexcel 考试中会丢分。
9. Worked Example from Mixed Exercise 8 | 混合练习8典型例题解析
Let' al term gives coefficient of x: n · b = 15. Coefficient of x²: C(n, 2) · b² = 90. Substitute C(n, 2) = n(n - 1)/2. So (n(n - 1)/2) · b² = 90. Express b = 15/n, then (n(n - 1)/2) · (225/n²) = 90. Simplify: (n - 1)/n · 112.5 = 90?
让我们逐步解析一个典型的混合练习8题目：(1 + bx)ⁿ 的展开式前三项为 1 + 15x + 90x²。求 n 和 b。解：通项给出 x 的系数：n · b = 15。x² 的系数：C(n, 2) · b² = 90。代入 C(n, 2) = n(n - 1)/2，得 (n(n - 1)/2) · b² = 90。由 b = 15/n 代入，得 (n(n - 1)/2) · (225/n²) = 90。化简：(n - 1)/n · 112.5 = 90？稍不精确，应保持分数形式。
5/5 = 3. So (1 + 3x)⁵ produces the given terms. Always check: C(5,1)·3=15, C(5,2)·3²=10·9=90. Correct.
更好的做法：保持分数， = 90 → (n(n - 1)/2) · (225/n²) = 90 → (n - 1)/n · (225/2) = 90 → (n - 1)/n = 180/225 = 4/5。于是 5(n - 1) = 4n → 5n - 5 = 4n → n = 5。然后 b = 15/5 = 3。因此 (1 + 3x)⁵ 生成所给项。永远要检验：C(5,1)·3=15，C(5,2)·3²=10·9=90，正确。
This structured approach – writing general terms, forming equations, substituting carefully, and simplifying – will solve any constant‑finding problem in the mixed exercise. Practise identifying whether the given term is the first, second or third term to avoid off‑by‑one errors.
这种结构化的方法——写出通项、列出方程、仔细代入并进行化简——可以解决混合练习中任何求常数的问题。多加练习识别所给项是第一项、第二项还是第三项，以避免索引偏移错误。
10. Summary and Exam Preparation Tips | 总结与备考建议
Mastery of Mixed Exercise 8 comes down to three core skills: expanding (a + b)ⁿ confidently, linking coefficients to the general term, and interpreting problems with unknown constants. Practise with both pure expansion and applied approximations. In the exam, show the general term formula explicitly and write coefficients before substituting numbers to minimise arithmetic errors.
掌握混合练习8归结为三项核心技能：自信地展开 (a + b)ⁿ、将系数与通项联系起来，以及解读含未知常数的问题。既要练习纯展开，也要练习应用近似。在考试中，明确写出通项公式，先以代数形式写出系数再代入数字，以尽量减少算术错误。
Time yourself on mixed exercises to build speed, and always review the official Edexcel mark schemes to see how working is awarded. Remember that coefficients must be fully simplified, and the final answer must be clearly indicated. With systematic preparation, binomial expansion can become one of your strongest topics.
为混合练习计时以提高速度，始终查阅官方 Edexcel 评分方案以了解步骤分的分配方式。切记系数必须完全化简，最终答案须明确标示。通过系统化准备，二项式展开完全可以成为你最拿手的专题之一。
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