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9.4: 合理化代数分数

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    171281

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    定义：有理表达式的分母

    如果有理表达式的分母包含涉及激进的总和或差，则最好始终通过将分子和分母乘以分母的共轭来合理化分母。

    分母的共轭包含相同的项，但运算相反（加法或减法）。

    示例 Template:Index

    合理化分母并简化：

    1. \(\dfrac{1}{1 − \sqrt{x}}\)
    2. \(\dfrac{1}{\sqrt{x} − \sqrt{y}}\)
    3. \(\dfrac{\sqrt{x} + \sqrt{y}}{\sqrt{x} − \sqrt{y}}\)

    解决方案

    1. \(\begin{array} &&\dfrac{1}{1 − \sqrt{x}} &\text{Example problem} \\ &\dfrac{(1)(1 + \sqrt{x})}{(1 − \sqrt{x})(1 + \sqrt{x})} &\text{Multiply both numerator and denominator by the conjugate, which is \((1+\sqrt{x})\)}\\ &\ dfrac {1 +\ sqrt {x}} {1 −\ sqrt {x} +\ sqrt {x} − (\ sqrt {x}) ^2} &\ text {FOIL 分母。}\\ &\ dfrac {1 +\ sqrt {x}} {1 −\ 取消 {\ sqrt {x}} − (\ sqrt {x}) ^2} &\ text {移除总和为零的相反项。}\\ &\ dfrac {1 +\ sqrt {x}} {1 − x} &\ text {方块的根\(x\)，数量平方为\(x\)。}\\ &\ dfrac {1 +\ sqrt {x}} {1 − x} &\ text {分母合理化后的最终答案，这意味着分母中没有平方根项。} \ end {array}\)
    1. \(\begin{array} &&\dfrac{1}{\sqrt{x} − \sqrt{y}} &\text{Example problem} \\ &\dfrac{(1)(\sqrt{x} + \sqrt{y})}{(\sqrt{x} − \sqrt{y})(\sqrt{x} + \sqrt{y})} &\text{Multiply both numerator and denominator by the conjugate, which is \((\sqrt{x} + \sqrt{y})\)}\\ &\ dfrac {(\ sqrt {x} +\ sqrt {y})} {(\ sqrt {x}) ^2 −\ sqrt {x}\ sqrt {y} +\ sqrt {y} − (\ sqrt {y}) ^2} &\ text {FOIL 分母。}\\ &\ dfrac {(\ sqrt {x} +\ sqrt {y})} {(\ sqrt {x}) ^2 −\ 取消 {\ sqrt {x}\ sqrt {y}} +\ cancel {\ sqrt {y}} − (\ sqrt {y}) ^2} &\ text {删除相反的术语总和为零。}\\ &\ dfrac {\ sqrt {x} +\ sqrt {y}} {x − y} &\ text {数量的平方根为\(x\)，数量平方为\(y\)，平方根为\(y\)。}\\ &\ dfrac {\ sqrt {x} +\ sqrt {y} {x} {x − y}} {x − y} &\ text {合理化了，意思是\(x\)分母中没有平方根项。} \ end {array}\)
    1. \(\begin{array} &&\dfrac{\sqrt{x} + \sqrt{y}}{\sqrt{x} − \sqrt{y}} &\text{Example problem} \\ &\dfrac{(\sqrt{x} + \sqrt{y})(\sqrt{x} + \sqrt{y})}{(\sqrt{x} − \sqrt{y})(\sqrt{x} + \sqrt{y})} &\text{Multiply both numerator and denominator by the conjugate, which is \((\sqrt{x} + \sqrt{y})\)}\\ &\ dfrac {(\ sqrt {x}) ^2 + 2 (\ sqrt {x}\ sqrt {y}) + (\ sqrt {y}) ^2 − (\ sqrt {x}) ^2 −\ sqrt {y} − (\ sqrt {y}) ^2} &\ text {FOIL 分子和分母。}\\ &\ dfrac {x + 2\ sqrt {x}\ sqrt {y}\ sqrt {y} {(\ sqrt {x}) ^2 −\ 取消 {\ sqrt {x}\ sqrt {y}} − (\sqrt {y}) ^2} &\ text {移除总和为零的相反项。}\\ &\ dfrac {x + 2\ sqrt {x}\ sqrt {y} {x − y} {x − y} &\ text {x}\ sqrt {y} + y} {x − y} &\\(x\)\(x\)\(y\)\(y\)text {对分母进行了合理化的最终答案，这意味着分母中没有平方根项。} \ end {array}\)
    练习 Template:Index

    合理化分母并简化：

    1. \(\dfrac{x}{1 − \sqrt{x}}\)
    2. \(\dfrac{1}{1 − \sqrt{x}}\)
    3. \(\dfrac{2 \sqrt{x}}{\sqrt{x} − 1}\)
    4. \(\dfrac{x − 1}{\sqrt{x} − 1}\)