初中數學（香港課程）/多項式的因式/3

小學的分數加減：

• ${\frac {1}{5}}+{\frac {2}{5}}={\frac {3}{5}}$
• ${\frac {3}{5}}-{\frac {1}{5}}={\frac {2}{5}}$

有相同分母代數分式的加減運算只需分子加減

示範例子1 化簡 ${\frac {9}{l-6}}-{\frac {4}{l-6}}$ ${\frac {8w}{w+7}}+{\frac {56}{w+7}}$ 解 ${\frac {9}{\color {red}l-6}}-{\frac {4}{\color {red}l-6}}={\frac {5}{l-6}}$ ${\begin{aligned}{\frac {8w}{w+7}}+{\frac {56}{w+7}}&={\frac {8w+56}{w+7}}\\&={\frac {8(w+7)}{w+7}}=8\end{aligned}}$

同類習題1 {{{3}}}

小學的分數加法（通分母）：

• ${\frac {1}{4}}+{\frac {1}{6}}={\frac {3}{12}}+{\frac {2}{12}}={\frac {5}{12}}$

分母不同的分式需要把全部分式化為一樣分母，通常會化為全部分母的最少公倍數

示範例子2 化簡 ${\frac {3}{x}}-{\frac {1}{2x}}$ ${\frac {4}{5b-6}}+{\frac {7}{20b-24}}$ ${\frac {10}{n-8}}+{\frac {9}{8-n}}$ 解 ${\begin{aligned}{\frac {3}{x}}-{\frac {1}{2x}}&={\frac {3\cdot 2}{x\cdot 2}}-{\frac {1}{2x}}\\&={\frac {6}{2x}}-{\frac {1}{2x}}={\frac {5}{2x}}\end{aligned}}$ ${\begin{aligned}{\frac {4}{5b-6}}+{\frac {7}{20b-24}}&={\frac {16}{4(5b-6)}}+{\frac {7}{4(5b-6)}}\\&={\frac {23}{4(5b-6)}}\end{aligned}}$ ${\begin{aligned}{\frac {10}{n-8}}+{\frac {9}{8-n}}&={\frac {10}{n-8}}-{\frac {9}{n-8}}\\&={\frac {1}{n-8}}\end{aligned}}$

同類習題2 {{{3}}}

示範例子3 化簡 ${\frac {4}{9a}}+{\frac {5}{6a}}-{\frac {7}{8a}}$ ${\frac {1}{30y-20}}-{\frac {8}{18-27y}}$ 解 ${\begin{aligned}{\frac {4}{9a}}+{\frac {5}{6a}}-{\frac {7}{8a}}&={\frac {32}{72a}}+{\frac {60}{72a}}-{\frac {63}{72a}}\\&={\frac {29}{72a}}\end{aligned}}$ ${\begin{aligned}{\frac {1}{30y-20}}-{\frac {8}{18-27y}}&={\frac {1}{10(3y-2)}}-{\frac {8}{9(2-3y)}}\\&={\frac {1}{10(3y-2)}}+{\frac {8}{9(3y-2)}}\\&={\frac {9}{90(3y-2)}}+{\frac {80}{90(3y-2)}}={\frac {89}{90(3y-2)}}\end{aligned}}$

同類習題3 {{{3}}}

示範例子4 化簡： $4-{\frac {7}{m+1}}$ ${\frac {3}{5z+1}}+{\frac {2}{z-6}}$ 解 ${\begin{aligned}4-{\frac {7}{m+1}}&={\frac {4(m+1)}{m+1}}-{\frac {7}{m+1}}\\&={\frac {4m+4}{m+1}}-{\frac {7}{m+1}}={\frac {4m-3}{m+1}}\end{aligned}}$ ${\begin{aligned}{\frac {3}{5z+1}}+{\frac {2}{z-6}}&={\frac {3(z-6)}{(5z+1)(z-6)}}+{\frac {2(5z+1)}{(5z+1)(z-6)}}\\&={\frac {3z-18}{(5z+1)(z-6)}}+{\frac {10z+2}{(5z+1)(z-6)}}\\&={\frac {13z-16}{(5z+1)(z-6)}}\end{aligned}}$

同類習題4 {{{3}}}