初中數學（香港課程）/多項式的因式/2

由多項式P除以多項式Q組成不是多項式的代數式（$\frac{P}{Q}$）稱作代數分式（英：Algebraic Fraction）

• 若果Q是數字，則成為多項式（$Q\ne 0$），代數分式的分母不可以是數字
• 例子：$\frac{4}{x}$、$\frac{a+1}{a-3}$、 $\frac{1}{p^2-5pq+q^2}$

Extended content
$\frac{x}{9x}$和$\frac{7a-56}{a-8}$能否直接化簡？如是，試化簡 試因式分解$\frac{7a-56}{a-8}$的分子，並化簡

代數分式中分子和分母的公因式需因式分解才可取出，之後再消去公因式化簡

示範例子1 化簡： $\frac{9b-54}{b^2-6b}$ $\frac{20y}{4y+8}$ $\frac{3c^3+30c^{2}d}{7c^{2}d+70c{d}^2}$ 解 $\begin{array}{cc}\frac{9b-54}{b^2-6b}& =\frac{9(b-6)}{b(b-6)}\\ & =\frac{9}{b}\end{array}$ $\begin{array}{cc}\frac{20y}{4y+8}& =\frac{20y}{4(y+2)}\\ & =\frac{5y}{y+2}\end{array}$ $\begin{array}{cc}\frac{3c^{2}d+30c{d}^2}{7c^3+70c^{2}d}& =\frac{3cd(c+10d)}{7c^2(c+10d)}\\ & =\frac{3d}{7c}\end{array}$

同類習題1 {{{3}}}

示範例子2 化簡： $\frac{8-z}{6z-48}$ $\frac{9r+45-rs-5s}{r+5}$ 解 $\begin{array}{cc}\frac{8-z}{6z-48}& =\frac{-(z-8)}{6(z-8)}\\ & =-\frac{1}{6}\end{array}$ $\begin{array}{cc}\frac{s-9}{9r+45-rs-5s}& =\frac{-(9-s)}{9(r+5)-s(r+5)}\\ & =\frac{-(9-s)}{(9-s)(r+5)}\\ & =-\frac{1}{r+5}\end{array}$

同類習題2 {{{3}}}

乘法與除法

小學的分數乘法： $\frac{5}{6}\cdot \frac{8}{15}=\frac{5^1}{6_3}\cdot \frac{8^4}{{15}_3}=\frac{4}{9}$

代數分式的乘法與分數步驟相同，只需把兩個分式的分子互乘、分母互乘即可，分子、分母之間可約簡

• 化簡仍需因式分解

小學的分數除法： $\frac{\frac{1}{14}}{\frac{3}{7}}=\frac{1}{14}\cdot \frac{7}{3}=\frac{1}{6}$

同樣地，代數分式的除法中，分母和分子需倒轉，化為乘法

示範例子3 化簡： $\frac{2k+18}{10}\cdot \frac{8}{k^2+9k}$ $\frac{\frac{21u^5}{6v^4}}{\frac{7u^3}{24v}}$ $\frac{\frac{4n+20}{n-1}}{\frac{3n+15}{n^2-n}}$ 解 $\begin{array}{cc}\frac{2k+18}{10}\cdot \frac{8}{k^2+9k}& =\frac{\cancel{2}\cancel{(k+9)}}{{\cancel{10}}_5}\cdot \frac{8}{k\cancel{(k+9)}}\\ & =\frac{8}{5k}\end{array}$ $\begin{array}{cc}\frac{\frac{21u^5}{6v}}{\frac{7u^3}{24v^4}}& =\frac{21u^5}{6v}\cdot \frac{24v^4}{7u^3}\\ & =\frac{12u^2}{v^3}\end{array}$ $\begin{array}{cc}\frac{\frac{4n+20}{n-1}}{\frac{3n+15}{n^2-n}}& =\frac{4n+20}{n-1}\cdot \frac{n^2-n}{3n+15}\\ & =\frac{4(n+5)}{n-1}\cdot \frac{n(n-1)}{3(n+5)}\\ & =\frac{4n}{3}\end{array}$

同類習題3 {{{3}}}

示範例子4 化簡： $\frac{56m-63m^2}{m^3}\cdot \frac{m^5}{42-54m}$ $\frac{\frac{4h+8}{h^6}\cdot h}{\frac{h^2+2h}{h^3}}$ 解 $\begin{array}{cc}\frac{56m-72m^2}{m^3}\cdot \frac{m^5}{42-54m}& =\frac{8m(7-9m)}{m^3}\cdot \frac{m^5}{6(7-9m)}\\ & =\frac{4}{3m^3}\end{array}$ $\begin{array}{ccc}\frac{\frac{4h+8}{h^6}\cdot h}{\frac{h^2+2h}{h^3}}& =\frac{4h+8}{h^2}\cdot h^6\cdot \frac{h^3}{h^2+2h}& =\frac{4(h+2)}{h^5}\cdot \frac{h^3}{h(h+2)}\\ & =\frac{4}{h^3}\end{array}$

同類習題4 {{{3}}}