🧩 因式分解：把式子拆回乘法🧩 Factoring: Break It Back into a Product
上一片里，我们把 (x+2)(x+3) 乘开，拼成了 x²+5x+6。这一片反过来：拿着拼好的式子，怎么把它拆回两个括号相乘？这就是因式分解。Last leaf we multiplied (x+2)(x+3) out and built x²+5x+6. This leaf goes the other way: given the built-up expression, how do we split it back into two brackets? That is factoring.
第 1 步：拆分，就是乘法的倒放Step 1: Splitting is multiplication run backwards
先看更小的例子——数字 12 能怎么拆？答案不止一组。Start smaller — how can the number 12 be split? There is more than one answer.
拆积木和拆数字是同一件事：12 = 3 × 4；x²+5x+6 = (x+2)(x+3)。拆出来的两块再乘回去，必须得到原来的式子——这就是"逆操作"。Splitting blocks is the same job as splitting numbers: 12 = 3 × 4; x²+5x+6 = (x+2)(x+3). Multiply the two parts back together and you must get the original — splitting is an inverse operation.
那 6x+9 这种没有两个括号的式子，从哪下手？So where do we start with something like 6x+9, which shows no brackets at all?
第 2 步：先拿走公共零件（提公因式）Step 2: Take out the shared part first
每一项都含有的零件，先提到括号外面：6x+9 = 3(2x+3)。找公共零件看三处：数字系数都能被 3 整除、相同的字母、字母的最低次——都能提，就一路提到最简。Whatever every term shares, pull it outside the bracket first: 6x+9 = 3(2x+3). Check three places — the number parts (both divide by 3), the shared letters, and the lowest power — and keep pulling until nothing is left to pull.
那 x²+5x+6 这种三项式，怎么找到那两个括号？And how do we find the two brackets for a three-term expression like x²+5x+6?
第 3 步：十字相乘——二次三项式的钥匙Step 3: The cross grid — the key to quadratics
把 x² 拆成 x·x，把 6 拆成 2·3；再交叉相加：2x+3x = 5x，恰好是中间项——配对成功，得到 (x+2)(x+3)。如果对不上，就换一种拆法再试。Split x² into x·x and 6 into 2·3; then cross-add: 2x+3x = 5x, exactly the middle term — a match, giving (x+2)(x+3). If it does not match, try a different split.
🎮 你来拆积木（30 秒）🎮 Your turn: split the blocks (30 seconds)
道理讲完了。4 块积木等着你：每一关选出正确的拆法，拆到最简。Theory done. Four blocks await: pick the correct split in every level and strip each one down to its simplest form.
一句话记住它：因式分解 = 乘法的倒放。先提公因式，二次三项式用十字相乘，拆完乘回去验算。Remember it in one line: factoring is multiplication run backwards. Take out the common factor, try the cross grid for quadratics, and multiply back to check.
拆分是逆操作：12=3×4，x²+5x+6=(x+2)(x+3)，乘回去能验算Splitting is an inverse: 12=3×4 and x²+5x+6=(x+2)(x+3) multiply back to the original提公因式：每一项都有的零件先拿走，如 6x+9=3(2x+3)Common factor first: pull out what every term shares, as in 6x+9=3(2x+3)十字相乘：拆两头、交叉加，和等于中间项就配对成功Cross grid: split the ends, cross-add — matching the middle term means a match
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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