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9.1: 简化有理表达式

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    171295

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    定义：理性表达

    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。

    示例 Template:Index

    简化有理表达式：

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

    解决方案

    1. \(\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}\)
    1. \(\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}\)
    1. \(\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}\)
    练习 Template:Index

    简化有理表达式：

    1. \(\dfrac{2x^2 + 3x − 2}{2x^2 + 5x − 3}\)
    2. \(\dfrac{(t^2 + 4)(2t − 4) − (t^2 − 4t + 4)(2t)}{(t^2 + 4)^2}\)
    3. \(\dfrac{(2)(x − 4)(x^2 + 4x + 4)}{(x + 2)(x^2 − 16)}\)
    4. \(\dfrac{12x^2 + 19x − 21}{12x^2 + 38x − 40}\)