9.1: 简化有理表达式

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有理表达式被写成多项式的商

$\dfrac{P(x)}{Q(x)} \nonumber$

其中 $P(x)$ 和 $Q(x)$ 是一个变量中的多项式 $x$ 。

要简化有理表达式，请同时对分子和分母进行分数，并从分子和分母中移除公共因子。简化的有理表达式只有一个除法以及一个分子和分母。如果无法对表达式进行分解，则无法简化有理表达式。

简化有理表达式：

$\dfrac{x^2 + 2x − 3}{x^2 + 4x + 3}$
$\dfrac{(x^2 + 1)^2 (−2) + (2x)(2)(x^2 + 1)(2x)}{(x^2 + 1)^4}$
$\dfrac{(x^2 + 1) \frac{1}{2} (x^{−\frac{1}{2}}) − (2x)(x^{\frac{1}{2}})}{(x^2 + 1)^2}$

解决方案

$\begin{array} &&\dfrac{x^2 + 2x − 3}{x^2 + 4x + 3} &\text{Example problem} \\ &\dfrac{(x + 3)(x − 1)}{(x + 3)(x + 1)} &\text{Factor both numerator and denominator.} \\ &\dfrac{\cancel{(x + 3)}(x − 1)}{\cancel{(x + 3)}(x + 1)} &\text{Remove common factors, because $\dfrac{x + 3}{x + 3} = 1$ }\\ &\ dfrac {x − 1} {x + 1} &\ text {最终答案}\ end {array}$

$\begin{array} &&\dfrac{(x^2 + 1)^2 (−2) + (2x)(2)(x^2 + 1)(2x)}{(x^2 + 1)^4} &\text{Example problem} \\ &\dfrac{2(x^2 + 1)[(x^2 + 1)(−1) + (2x)(2x)]}{(x^2 + 1)^4} &\text{Factor out 2(x^2 + 1)} \\ &\dfrac{2 \cancel{(x^2 + 1)}[(x^2 + 1)(−1) + (2x)(2x)]}{\cancel{(x^2+1)}(x^2 + 1)^3} &\text{Remove common factors, because $\dfrac{x^2 + 1}{x^2 + 1} = 1$ }\\ &\ dfrac {2 [−x^2 − 1 + 4x^2]} {(x^2 + 1) ^3} &\ text {通过类似术语相乘和合并来简化}\\ &\ dfrac {2 (3x^2 + 1) ^3} &\ text {最终答案}\ end {array}$

$\begin{array} &&\dfrac{(x^2 + 1) \frac{1}{2} (x^{−\frac{1}{2}}) − (2x)(x^{\frac{1}{2}})}{(x^2 + 1)^2} &\text{Example problem} \\ &\dfrac{\frac{(x^2+1)}{2x^{\frac{1}{2}}} − (2x)(x^{ \frac{1}{2} })}{(x^2 + 1)^2} &\text{Work with the negative exponent in the first term of the numerator by moving the factor to the denominator of the first term, next to the $2$ .}\\ &\ dfrac {(x^2 + 1) − (2x) (x^ {\ frac {1} {2}}) 2 (x^ {\ frac {1} {2}}) {\ dfrac {1} {2}} {(x^2 + 1) ^2}} &\ text {公分母}\\ &\ dfrac {x^2 + 1 − 4x^2} {(2x^ {\ frac {1} {2}}) (x^2 + 1) ^2} &\ text {通过类似术语相乘和合并来简化}\\ &\ dfrac {−3x^2 + 1} {(2x^ {\frac {1} {2}}) (x^2 + 1) ^2} &\ text {最终答案}\ end {array}$

简化有理表达式：