Edexcel GCSE Algebra 展开括号 因式分解 数列

Edexcel GCSE Algebra 展开括号 因式分解 数列

Edexcel 的 GCSE Mathematics（考纲编号 1MA1）把 algebra 单独列成一个主题，三张试卷每张 80 分、各 1 小时 30 分，Paper 1 不允许用计算器。代数在每张卷子上都出现，从展开括号、化简这类一两行的短题开始，一直走到因式分解、联立方程和数列。这篇按 1MA1 的编排顺序把这条线走一遍，Foundation 与 Higher 的差别单独标注。

Edexcel GCSE Mathematics (specification 1MA1) gives algebra its own topic area. Each of the three papers lasts 1 hour 30 minutes and carries 80 marks, and Paper 1 is the one where calculators are not allowed. Algebra appears on all three papers, starting with short expanding and simplifying questions and running through factorising, simultaneous equations and sequences. What follows copies the order of the 1MA1 specification, and anything that belongs to the Higher tier only is labelled as such.

展开括号与合并同类项

化简题的第一步永远是看清哪些项是同类项：3x 与 5x 能合并，3x 与 3x² 不能合并。展开括号时每一项都要乘到，单括号用分配律，双括号 (x+3)(x-5) 一共四个乘积，写成 x²-5x+3x-15 再收成 x²-2x-15。很多学生以为 (2x+1)² 等于 4x²+1，正确答案是 4x²+4x+1，中间那一项来自 2 乘 2x 乘 1。负号是第二个高发错误，(3x-2)(x+4) 里 -2 乘 4 得 -8，符号写反整题就废了。三个括号连乘的题只是分两次做，(x+1)(x+2)(x+3) 先做前两个得 x²+3x+2，再与第三个相乘，多写一行草稿比心算稳。

Like terms are the first thing to look for: 3x and 5x combine, while 3x and 3x squared do not. Expanding a single bracket uses the distributive law; expanding two brackets produces four products, which are then collected. Squaring is where marks disappear most often, because the middle term is twice the product of the two terms and not simply the square of each one. Sign slips on the second bracket of a product sit right behind it, and one misplaced minus usually takes the whole answer down with it.

因式分解的三条主线

拿到一个要分解的式子，先看有没有公因式，6x²+9x 提出 3x 得 3x(2x+3)，这一步没做完后面全乱。没有公因式就看平方差，x²-9 写成 (x+3)(x-3)，4x²-25 写成 (2x+5)(2x-5)。剩下的二次三项式找两个数，乘积等于常数项、和等于一次项系数，x²+7x+12 的两个数是 3 和 4，写成 (x+3)(x+4)。Higher 会出现 x² 系数不是 1 的式子，例如 6x²+11x-35，用十字相乘或先分组再提公因式，1MA1 考纲把这类题放在 Higher 内容里。分解完把两个括号乘回去验算一遍，几秒钟就能确认中间项对不对。

Factorising follows a fixed order of checks, and the order matters more than the tricks. A common factor comes first, since 6x²+9x turns into 3x(2x+3) and skipping that step makes every later step harder. If nothing is common to all the terms, test for a difference of two squares. For a quadratic with a leading coefficient of 1, look for two numbers whose product is the constant term and whose sum is the coefficient of x. When the coefficient of x squared is not 1, Higher tier questions need a grid method or grouping, and the specification lists those expressions as Higher only.

一元一次方程的去分母顺序

解 (x+1)/3-(x-2)/2=1 这类题，第一步是两边同乘 6 把分母清掉，写成 2(x+1)-3(x-2)=6，右边那个 1 也要乘。接着展开、移项、合并、系数化 1，得到 x=2。常见的错误是只把左边两项的分母乘掉，右边原样留下，答案就此偏掉。系数是分数但分母里没有未知数的方程，处理方式一样，3(x-4)=2x+5 展开得 3x-12=2x+5，移项后 x=17，代回去两边都等于 39。两边都有括号的题先各自展开再移项，5(x-1)=3(x+4) 展开成 5x-5=3x+12，移项得 2x=17，x=8.5，答案允许是小数或分数，不必硬凑整数。做完代回原式检查一次，两秒钟的事，能抓住大部分符号错误。

Clearing the fractions is the opening move whenever an equation carries denominators, and the multiplier has to be applied to every term, including the lonely constant on the right hand side. After that the work is routine: expand, collect, divide by the coefficient of x. Putting the answer back into the original equation takes a few seconds and catches most sign errors before they cost anything.

联立方程：加减消元与代入

两个未知数的方程组看系数决定方法。3x+2y=12 与 x-2y=4 相加就能消掉 y，得到 4x=16，x=4，代回第二条得 4-2y=4，所以 y=0，检验时两个方程都要对。2x+y=7 与 x²+y=10 这种一条直线配一条曲线的组合只能用代入，把 y=7-2x 塞进第二条式子得到 x²+7-2x=10，整理成 x²-2x-3=0 再分解成 (x-3)(x+1)=0。Higher 卷子上后一种每年都会出现，答案通常两组，两个都要写出来，只写一个按未完成给分。

Which method to use depends on the coefficients in front of the variables. When the coefficients of one variable are equal or opposite in the two equations, adding or subtracting removes it in a single line. When one equation is linear and the other contains a square, substitution is the only route: rearrange the linear equation for one variable, feed it into the other equation, and solve the quadratic that comes out. Higher tier mark schemes expect both solutions, usually as coordinate pairs.

不等式与数轴上的开闭点

不等式和解方程差在一个地方：两边同乘或同除负数，不等号方向要反过来。-2x>6 除完得到 x<-3，忘了变号是这一块最常见的失分点。答案画在数轴上时，端点是空圈还是实心圈取决于原题有没有等号。3(x-2) 小于等于 12 这类题先展开得 3x-6 小于等于 12，再解得 x 小于等于 6，题目若问整数解，就要把 6 以内的整数一个个列出来。两端都有未知数的不等式先把 x 集中到一边，5x-3 小于 2x+9 移项得 3x 小于 12，所以 x 小于 4，和方程的唯一区别仍是最后一步要不要变号。Higher 还会要求用集合记号写解集，例如 x<-3 写成 {x : x < -3}。

Inequalities behave like equations with one extra rule attached: multiplying or dividing both sides by a negative number flips the sign. On a number line, a strict inequality gets an open circle and an inequality that allows equality gets a filled one. Set notation, such as an expression inside braces with a colon, is a Higher tier requirement on 1MA1.

数列与第 n 项的求法

差相等的数列是线性数列，第 n 项写成 dn 加常数，5, 8, 11, 14 的公差是 3，第 n 项就是 3n+2，第 20 项代进去是 62。二阶差相等的数列是二次数列，Higher 要求写出含 n² 的表达式，2, 7, 16, 29 的二阶差是 4，n² 的系数是二阶差的一半也就是 2，再减去 2n² 与原数列的差得到修正项。反过来判断某个数在不在数列里，把它当成方程解，解出正整数才算在数列里，比如 100 在 3n+2 里解不出整数，就不在这一列中。

Sequences in the 1MA1 specification fall into two families. A constant first difference gives a linear nth term, so a sequence with a common difference of 3 has an expression of the form 3n plus a constant. A constant second difference gives a quadratic nth term, which is Higher tier work: halve the second difference to get the coefficient of n squared, subtract that sequence from the original, then fit the remaining linear part. Testing whether a given number belongs to a sequence uses the same skill backwards, and only whole number solutions count.

直线方程、斜率与垂直关系

直线写成 y=mx+c，m 是斜率，c 是 y 轴截距。给两点求方程，先算斜率，再代一个点求 c。平行线斜率相同，垂直的两条线斜率乘积是 -1，这一条只出现在 Higher。求过点 (4,1) 且垂直于 y=2x+3 的直线，斜率是 -1/2，代入得到 y=-1/2x+3。水平线写成 y=3，斜率是 0，竖直的线写成 x=2，斜率不存在，这两种在图像题里经常被要求直接写出来。读图题里斜率常被读成截距，画图时先标截距再走斜率能少出错。

A straight line written as y=mx+c carries its gradient in m and its intercept on the y axis in c. From two points, work out the gradient first and substitute one point afterwards to pin down the intercept. Parallel lines share the same gradient, while the gradients of perpendicular lines multiply to -1, which is a Higher tier rule. Graph reading questions sometimes confuse the gradient with the intercept, so marking the intercept before stepping out the gradient keeps the sketch honest.

二次函数图像、配方与判别式

二次函数的图像是抛物线，x² 的系数为正就开口向上。配方把一般式变成 a(x+p)²+q 的形式，顶点直接读出来，x²-6x+5 配成 (x-3)²-4，顶点是 (3,-4)。判别式 b²-4ac 决定图像与 x 轴的交点个数：大于 0 有两个，等于 0 有一个，小于 0 没有，图像整条在 x 轴上方或下方。解一元二次方程时能分解就分解，x²-5x+6=0 分解成 (x-2)(x-3)=0，根是 2 和 3。不能分解就用求根公式，配方还能直接读出最值：x² 的系数为正时括号里的平方最小取 0，函数的最小值就是 q，求最大面积一类题会用到这个结论。Edexcel 的 1MA1 会把公式印在试卷的公式表上，考生不必背，但要会代。

Quadratic graphs are parabolas, opening upwards when the coefficient of x squared is positive. Completing the square rewrites the expression in a form that shows the turning point directly. The discriminant, b squared minus 4ac, settles how many times the curve meets the x axis: two crossings, one, or none at all. Choosing between factorising and the quadratic formula is a matter of inspection, and the formula itself sits on the formulae sheet that Pearson Edexcel supplies with 1MA1 papers, so the skill being tested is substitution rather than memory.

列方程解几何题与实际问题

代数在几何题里通常以列式的方式出现：长方形周长 26、长比宽多 3，设宽为 x，长就是 x+3，方程 2(x+x+3)=26，解得 x=5，长 8。面积题同理，面积 40、长比宽多 3，得到 x(x+3)=40，整理成 x²+3x-40=0，分解成 (x+8)(x-5)=0，长度只取正根 x=5。很多学生算出两个根就把两个都写上，长度不会为负，筛根这一步省不得。

Algebra turns up inside geometry and word problems as an instruction to form an equation. A rectangle with perimeter 26 whose length is 3 more than its width gives 2(x + x + 3) = 26 once the width is called x. An area problem works the same way but ends in a quadratic, and only the positive root survives because a length cannot be negative. Dropping that filter is a common way to lose the final mark after doing all the algebra correctly.

代数证明与迭代法

证明题的开头是把要证的东西写成代数。证明任意两个连续奇数的平方差是 8 的倍数，设两数为 2n+1 与 2n+3，平方差展开得 8n+8，提取 8 就结束了。同一个套路可以处理 (3n+1)²-(3n-1)²，展开后剩 12n，所以它是 12 的倍数。证明两个连续偶数的积是 8 的倍数，设 2n 与 2n+2，乘积是 4n(n+1)，n 与 n+1 里必有一个偶数，整个乘积就带着因子 8。写证明不能只代数字举例，代三个数只说明那三个数成立。迭代法在 Higher 出现，题目给 xₙ₊₁ 的表达式和一个起始值，要求算到指定精度，每一步保留足够小数位，提前四舍五入答案就飘。

Proof questions begin by writing the claim in algebra rather than in words. Showing that the difference between the squares of consecutive odd numbers is a multiple of 8 means naming the numbers 2n+1 and 2n+3, expanding, and factorising out the 8. Substituting a few numbers proves nothing beyond those numbers. Iteration appears on Higher papers as a recurrence with a starting value, and the marks are usually lost by rounding too early rather than by the algebra itself.

Paper 1 非计算器卷的代数节奏

Paper 1 不给计算器，代数题的数字通常是可以手算的，遇到除不尽的小数就该怀疑前面算错了。分数、根号、负数乘方的运算在非计算器卷上比重更高，化简根号 48 这种题用 16 乘 3 拆开写 4√3 就够用。三张卷子的代数题分数分布相近，前面的短题两三分，后面的多步题五六分。多步题的分数按步骤给，写出展开或移项那一步就有分，空着最亏。每张卷子 90 分钟 80 分，平均一分一分钟略多一点，卡住的题先跳过去。答案写完后回头看一眼量级，代数题的解通常落在 -10 到 10 之间，算出 250 这种数多半是中途漏了除法。

Paper 1 is the non-calculator paper, so its algebra questions are built around numbers that work out by hand, and an awkward decimal is a signal that something went wrong earlier. Marks in multi-step questions are awarded for intermediate working, which means a correct expansion or a properly rearranged line still scores even when the final answer slips. With 80 marks in 90 minutes there is a little over a minute per mark, so leaving a stuck question and coming back rarely costs anything.

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