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第 11 週 · 多項式與餘式定理 · 第 5 節

有理函數（代數分式）的加減乘除

分子分母都是多項式的「分數」—— 運算規則和你小學學的分數一模一樣，關鍵只有一句：先因式分解，再約分／通分。配滿圖例從零講透。

▶ 第一關： 認識代數分式 + 約分（先分解再約）
numbers23numeratordenominatoralgebraic fractionx + 1x − 2same idea: top ÷ bottom
step 1: factor top & bottom(x−2)(x+2)(x+2)(x+2)→step 2: cancel the common factor (x+2)x − 2x + 2simplest
① factorise→② find common factor→③ cancel whole block
WRONG ✗x + 2x + 3cannot cancel x — it is a TERMOK ✓2·(x+3)(x+3)(x−3)(x+3) is a FACTOR → cancel
▶ 第二關： 乘除（分子×分子、除法翻轉相乘）
multiply: top×top, bottom×bottomab×cd=a × cb × dnumerators together, denominators together
cross-cancel the (x−2) before multiplyingx + 1x − 2×x − 2x + 3=x + 1x + 3(x−2) top cancels (x−2) bottom
only the fraction after ÷ is flippedA÷x − 1x²=A×x²x − 1A stays; the divisor becomes its reciprocal
▶ 第三關： 加減（通分到公分母 = LCM）
warm-up with numbers: make denominators match12+13→36+26=56common denominator 6 = LCM(2, 3)
1/(x−1) + 1/(x+1): scale each to the common denominator1x−1×(x+1)→x+1(x−1)(x+1)1x+1×(x−1)→x−1(x−1)(x+1)=2xx² − 1
denominatorsxx²→LCM = highest powerx²notx³
⚔ BOSS： 混合 ×÷ 與 +− 的綜合化簡
master plan:① factorise→② ×÷ flip & cancel+− use LCM→③ cancel again → simplest