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简化有理表达式介绍

介绍

\begin{aligned} \dfrac68&= \dfrac{2\cdot 3}{2\cdot 4}&&\small{\gray{\text{分解}}} \\\\ &= \dfrac{\tealD{\cancel{2}}\cdot 3}{\tealD{\cancel{2}}\cdot 4}&&\small{\gray{\text{约分}}} \\ \\ &= \dfrac{3}{4} &&\small{\gray{\text{简化}}} \end{aligned}

例题一: 简化有理式$\dfrac{x^2+3x}{x^2+5x}$start fraction, x, squared, plus, 3, x, divided by, x, squared, plus, 5, x, end fraction

start fraction, x, squared, plus, 3, x, divided by, x, squared, plus, 5, x, end fraction, equals, start fraction, x, left parenthesis, x, plus, 3, right parenthesis, divided by, x, left parenthesis, x, plus, 5, right parenthesis, end fraction

start fraction, x, left parenthesis, x, plus, 3, right parenthesis, divided by, start color #11accd, x, end color #11accd, start color #aa87ff, left parenthesis, x, plus, 5, right parenthesis, end color #aa87ff, end fraction

\begin{aligned}\dfrac{\tealD x(x+3)}{\tealD x(x+5)}&=\dfrac{\tealD {\cancel {x}}(x+3)}{\tealD{\cancel x}(x+5)}\\ \\ &=\dfrac{x+3}{x+5} \end{aligned}

start fraction, x, plus, 3, divided by, x, plus, 5, end fraction ，其中x, does not equal, 0

关于等价表达式的说明

start fraction, x, squared, plus, 3, x, divided by, x, squared, plus, 5, x, end fraction$\quad$start fraction, x, plus, 3, divided by, x, plus, 5, end fraction，其中x, does not equal, 0

容易混淆的地方

start fraction, x, plus, 3, divided by, x, plus, 5, end fraction, space, space, start color #e07d10, does not equal, end color #e07d10 space, start fraction, 3, divided by, 5, end fraction

start fraction, start color #aa87ff, 2, end color #aa87ff, plus, 3, divided by, start color #aa87ff, 2, end color #aa87ff, plus, 5, end fraction, space, space, start color #e07d10, does not equal, end color #e07d10space, start fraction, 3, divided by, 5, end fraction

有理式简化步骤的总结

• 第一步: 对分子和分母进行因式分解。
• 第二步: 列出令有理式无意义的值。
• 第三步: 消去公因式。
• 第四步: 简化有理式，列出所有令原有理式无意义，但无法由其简化形式推断出来的值。

看看你的知识掌握地如何

1) 简化有理式 start fraction, 6, x, plus, 20, divided by, 2, x, plus, 10, end fraction

2) 简化有理式 start fraction, x, cubed, minus, 3, x, squared, divided by, 4, x, squared, minus, 5, x, end fraction.
，其中 x, does not equal

例题二: 简化有理式 $\dfrac{x^2-9}{x^2+5x+6}$start fraction, x, squared, minus, 9, divided by, x, squared, plus, 5, x, plus, 6, end fraction

start fraction, x, squared, minus, 9, divided by, x, squared, plus, 5, x, plus, 6, end fraction, equals, start fraction, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 3, right parenthesis, divided by, left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, plus, 3, right parenthesis, end fraction

start fraction, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 3, right parenthesis, divided by, start color #11accd, left parenthesis, x, plus, 2, right parenthesis, end color #11accd, start color #aa87ff, left parenthesis, x, plus, 3, right parenthesis, end color #aa87ff, end fraction

\begin{aligned}\dfrac{(x-3)\tealD{(x+3)}}{(x+2)\tealD{(x+3)}}&=\dfrac{(x-3)\tealD{\cancel{(x+3)}}}{(x+2)\tealD{\cancel{(x+3)}}}\\ \\ &=\dfrac{x-3}{x+2} \end{aligned}

start fraction, x, minus, 3, divided by, x, plus, 2, end fraction，其中x, does not equal, minus, 3

检查一下你对本课的理解

3) 简化有理式start fraction, x, squared, minus, 3, x, plus, 2, divided by, x, squared, minus, 1, end fraction.

4) 简化有理式start fraction, x, squared, minus, 2, x, minus, 15, divided by, x, squared, plus, x, minus, 6, end fraction
，其中 x, does not equal