📚 Products: Expanding, Factorising and Binomial Expansion | 乘积：展开、因式分解与二项式展开
In AQA A-Level Mathematics, the topic of products appears throughout pure mathematics: from expanding brackets and factorising polynomials to binomial expansions and rational expressions. This article reviews the essential methods you need to handle products confidently.
在 AQA A-Level 数学中，乘积这一主题贯穿纯数学：从去括号、因式分解到二项式展开和有理表达式。本文复习你从容应对乘积所需的基本方法。
1. What Does "Product" Mean? | “乘积”的含义
A product is the result of multiplying two or more algebraic factors together. In A-Level questions, you are often asked to expand a product, factorise a given expression into a product, or simplify a product of rational expressions.
乘积是将两个或多个代数因式相乘得到的结果。在 A-Level 题目中，你经常需要展开一个乘积、将给定表达式因式分解为乘积，或化简有理表达式的乘积。
For example, the product of (x + 2) and (x - 5) is written as (x + 2)(x - 5).
例如，(x + 2) 与 (x - 5) 的乘积写作 (x + 2)(x - 5)。
Always pay attention to the word "product" in exam questions because it tells you to multiply, not to add or subtract.
在考试题目中始终注意“乘积”一词，因为它提示你进行乘法，而不是加法或减法。
2. Multiplying Linear Factors | 线性因式相乘
When two linear factors are multiplied, each term in the first bracket is multiplied by each term in the second bracket. This is sometimes called the FOIL method for binomials.
当两个线性因式相乘时，第一个括号中的每一项都要与第二个括号中的每一项相乘。对于二项式，这有时称为 FOIL 法。
The general expansion is:
一般展开式为：
(x + a)(x + b) = x² + ax + bx + ab = x² + (a + b)x + ab
For example, (x + 3)(x - 2) = x² + 3x - 2x - 6 = x² + x - 6.
例如，(x + 3)(x - 2) = x² + 3x - 2x - 6 = x² + x - 6。
Do not forget to collect like terms after expanding, as this is often where marks are awarded in AQA papers.
展开后不要忘记合并同类项，因为 AQA 试卷中往往在这里给分。
3. Expanding Two Binomials | 两个二项式展开
For two general binomials, the product is found by multiplying every term in the first bracket by every term in the second bracket.
对于两个一般二项式，乘积通过将第一个括号中的每一项与第二个括号中的每一项相乘来求得。
(ax + b)(cx + d) = acx² + adx + bcx + bd
This simplifies to acx² + (ad + bc)x + bd.
它化简为 acx² + (ad + bc)x + bd。
Worked example: (2x + 1)(3x - 4) = 6x² - 8x + 3x - 4 = 6x² - 5x - 4.
例题：(2x + 1)(3x - 4) = 6x² - 8x + 3x - 4 = 6x² - 5x - 4。
Always check the signs when multiplying negative terms, because sign errors are the most common mistakes in expansion.
相乘含有负项时一定要检查符号，因为符号错误是展开中最常见的失误。
4. Expanding Three Binomials | 三个二项式展开
When three linear factors are multiplied, you should first multiply two of them, simplify, and then multiply the result by the third factor.
当三个线性因式相乘时，应先将其中的两个相乘并化简，然后再将结果与第三个因式相乘。
(x + a)(x + b)(x + c) = x³ + (a + b + c)x² + (ab + bc + ca)x + abc
1)(x - 2)(x + 3) = x³ + (1 - 2 + 3)x² + (-2 - 6 + 3)x + (1 × -2 × 3) = x³ + 2x² - 5x - 6.
例题： + 1)(x - 2)(x + 3) = x³ + (1 - 2 + 3)x² + (-2 - 6 + 3)x + (1 × -2 × 3) = x³ + 2x² - 5x - 6。
Expanding three brackets in one step is faster, but if you are unsure, expand two brackets first to reduce errors.
一步展开三个括号更快，但如果不确定，可以先展开两个括号以减少错误。
5. Special Products: Difference of Two Squares | 特殊乘积：平方差
A difference of two squares is a special product that must be recognised quickly because it appears frequently in AQA examinations.
平方差是一种特殊乘积，必须快速识别，因为它在 AQA 考试中频繁出现。
(a + b)(a - b) = a² - b²
For example, (x + 5)(x - 5) = x² - 25.
例如，(x + 5)(x - 5) = x² - 25。
It is called a difference because it is one square minus another. It does not work for a sum of squares: a² + b² has no real linear factors.
它之所以称为差，是因为它是一个平方减去另一个平方。它不适用于平方和：a² + b² 没有实数线性因式。
Recognising this pattern also helps with factorising expressions such as 4x² - 9, which becomes (2x + 3)(2x - 3).
识别这一模式还有助于因式分解诸如 4x² - 9 的表达式，它变为 (2x + 3)(2x - 3)。
6. Special Products: Perfect Squares | 特殊乘积：完全平方
A perfect square is the product of a binomial with itself, and it also has standard expansions that should be memorised.
完全平方是一个二项式与自身的乘积，它也有应当记住的标准展开式。
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
For example, (x + 4)² = x² + 8x + 16 and (3x - 2)² = 9x² - 12x + 4.
例如，(x + 4)² = x² + 8x + 16，而 (3x - 2)² = 9x² - 12x + 4。
or is to write (x + a)² = x² + a². The middle term 2ax is essential and cannot be omitted.
一个常见错误是把 (x + a)² 写成 x² + a²。中间项 2ax 是必需的，不能省略。
7. Factorising Quadratic Products | 二次乘积的因式分解
Factorising is the reverse process of expanding. For a monic quadratic of the form x² + bx + c, find two numbers that multiply to c and add to b.
因式分解是展开的逆过程。对于 x² + bx + c 形式的二次式，找出两个数，它们的乘积为 c，和为 b。
x² + bx + c = (x + p)(x + q)
where p + q = b and pq = c.
其中 p + q = b 且 pq = c。
xample: x² - 7x + 12 factors as (x - 3)(x - 4) because (-3) + (-4) = -7 and (-3)(-4) = 12.
例题：x² - 7x + 12 因式分解为 (x - 3)(x - 4)，因为 (-3) + (-4) = -7 且 (-3)(-4) = 12。
For non-monic quadratics such as 2x² + 5x - 3, use the ac method or trial and error to obtain (2x - 1)(x + 3).
对于 2x² + 5x - 3 这样的非首一二次式，使用 ac 法或尝试法得到 (2x - 1)(x + 3)。
8. Factorising Cubic Products | 三次乘积的因式分解
To factorise a cubic expression such as x³ + ax² + bx + c, first use the factor theorem to find one linear factor by testing small integer values of x.
要因式分解三次表达式，如 x³ + ax² + bx + c，先用因式定理通过代入小的整数值 x 来找到一个线性因式。
e one factor is found, divide the cubic by that factor to obtain a quadratic, then factorise the quadratic completely.
如果 f(p) = 0，则 (x - p) 是一个因式。找到一个因式后，将三次式除以该因式得到二次式，再将二次式完全因式分解。
a factor. Division gives x² - 5x + 6, and the full factorisation is (x + 1)(x - 2)(x - 3).
例题：f(x) = x³ - 4x² + x + 6。由于 f(-1) = 0，所以 (x + 1) 是一个因式。除法得到 x² - 5x + 6，完全因式分解为 (x + 1)(x - 2)(x - 3)。
Always state the fully factorised product as it is usually required for full marks.
始终写出完全因式分解后的乘积，因为题目通常要求完整答案才能得满分。
9. Binomial Expansion for Positive Integers | 二项式展开（正整数次幂）
The binomial theorem expands powers of a binomial without repeated multiplication. For a positive integer n, the expansion of (a + b)ⁿ is:
二项式定理可以在不重复相乘的情况下展开二项式的幂。对于正整数 n，(a + b)ⁿ 的展开为：
(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + ... + ⁿCₙ bⁿ
Here ⁿCᵣ = n! / [r!(n - r)!] is the binomial coefficient.
此处 ⁿCᵣ = n! / [r!(n - r)!] 是二项式系数。
For example, (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴.
例如，(1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴。
In AQA exams, you need to find a specific coefficient or write the expansion up to a required term, so practise using the formula rather than expanding manually.
在 AQA 考试中，你需要求出特定的系数或写出到指定项的展开式，因此要练习使用公式，而不是手动逐项展开。
10. Products of Rational Expressions | 有理表达式的乘积
To multiply rational expressions, factorise all numerators and denominators first, then cancel any common factors before multiplying across.
要乘以有理表达式，应先将所有分子和分母因式分解，然后在交叉相乘之前约去任何公因式。
(A/B) × (C/D) = (A × C) / (B × D)
: (x² - 4)/(x + 2) × (x + 3)/(x - 2) = [(x - 2)(x + 2)]/(x + 2) × (x + 3)/(x - 2) = x + 3.
例题：(x² - 4)/(x + 2) × (x + 3)/(x - 2) = [(x - 2)(x + 2)]/(x + 2) × (x + 3)/(x - 2) = x + 3。
Remember that cancellation is only valid for factors, not for individual terms. Also note any excluded values, such as x ≠ -2 and x ≠ 2 in this example.
记住约分只对因式有效，而不是对单独的项有效。还要注意任何排除值，例如在本例中 x ≠ -2 且 x ≠ 2。
11. Common Errors with Products | 乘积常见错误
Many students lose marks because they do not check their expansion, factorisation, or cancellation. The following table lists common errors and correct approaches.
许多学生因为没有检查展开、因式分解或约分而失分。下表列出常见错误与正确做法。
| Common error | Correct approach |
| (x + 2)² = x² + 4 | (x + 2)² = x² + 4x + 4 |
| Forgetting the minus sign when expanding (x - 3)(x + 2) | Expand as x² + 2x - 3x - 6 = x² - x - 6 |
| Cancelling terms instead of factors | Factorise first, then cancel whole factors |
| Missing middle term in binomial expansion | Use ⁿC₁ term or Pascal's triangle carefully |
test value, such as x = 1, into the original and expanded forms to verify they are equal.
始终代入一个测试值，例如 x = 1，到原式和展开式中检验它们是否相等。
12. Exam Technique for Product Questions | 乘积题考试技巧
In AQA A-Level Mathematics, product questions often appear as part of larger problems involving equations, inequalities, or calculus. Start by expanding or factorising neatly and systematically.
在 AQA A-Level 数学中，乘积题常常作为涉及方程、不等式或微积分的较大问题的一部分出现。开始时应整洁而有条理地展开或因式分解。
Show all steps of your multiplication and simplification, because method marks are available even if the final answer is incorrect.
展示所有相乘和化简步骤，因为即使最终答案错误，方法分也可以获得。
For guided questions, look at the target form given in the question. If the question asks for a product form, leave your answer as a product rather than expanding it.
对于有引导的题目，应查看题目给出的目标形式。如果题目要求乘积形式，就不要展开，而将答案保留为乘积。
Finally, check your answer by expanding a factorised result or substituting a simple value into both sides of an identity.
最后，通过展开因式分解结果或将一个简单值代入恒等式两边来检查答案。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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