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第 11 週 · 多項式與餘式定理 · 第 4 節

多項式的最大公因式 HCF 與最小公倍式 LCM

HCF / LCM 你小學在數字上早就學過（12 和 18）。多項式一模一樣 —— 只是把「質因數」換成「因式」。先分解，再比因式，秒懂。

▶ 第一關： 先複習數字的 HCF / LCM（12 與 18）
12 = 2² × 318 = 2 × 3²2 · 2 · 32 · 3 · 3HCF: lower power2 × 3 = 6LCM: higher power2² × 3² = 36
HCF = common,LOWER powervsLCM = all of them,HIGHER powerH → short (low)L → Large (high)
▶ 第二關： 多項式 HCF —— 先分解，取「共有」因式
numbers12 = 2² × 3prime factorsHCF / LCM of primespolynomialsx²−1 = (x−1)(x+1)factorsHCF / LCM of factorsfactor = the polynomial version of a prime
the 3-step method① factorizeeach polynomial→② HCFcommon, lower power→③ LCMall of them, higher power
x² − 1(x−1)(x+1)x² − 2x + 1(x−1)(x−1)common factor (x−1)HCF = (x − 1)
▶ 第三關： 多項式 LCM —— 取「所有」因式（韋恩圖）
three polynomials → HCF = 1, LCM = x(x−1)(x+1)x · (x−1)x · (x+1)(x−1) · (x+1)factors used:x(x−1)(x+1)no single factor is in all three → HCF = 1
Venn: factors of each polynomialx²−1x²−2x+1(x+1)extra (x−1)(x−1)shared∩ = HCF = (x−1)∪ = LCM = (x−1)²(x+1)
⚔ BOSS： 含系數 + 高次因式綜合 + 性質驗算
check:HCF×LCM=P × Q✓96 x³ (x−1)³ on both sides
next: LCM is the common denominator1x−1+1x+1→common denom =LCM (x−1)(x+1)