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9.2: 乘以有理表达式

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    171282

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    定义：乘以有理表达式

    要乘以有理表达式，请将分子表达式相乘并乘以分母表达式。 然后，如果可能的话，通过分解分子和分母并移除常见因子来进行简化。

    在使用\(4\)-项多项式时，尝试通过分组使用因子。

    注意：乘以有理表达式时不需要公分母！

    示例 Template:Index

    将有理表达式相乘：

    1. \(\dfrac{10t}{6t − 12} ∗ \dfrac{20t − 40}{30t^3}\)
    2. \(\dfrac{7x + 14}{ 2x^2 − 8} ∗ \dfrac{x^2 + 3x − 10}{14x + 21}\)
    3. \(\dfrac{3x^3 − 24}{2x^2 − 14x + 20} ∗ \dfrac{4x^3 − 20x^2 + 3x − 15}{3x^2 + 6x + 12}\)

    解决方案

    1. \(\begin{array} &&\dfrac{10t}{6t − 12} ∗ \dfrac{20t − 40}{30t^3} &\text{Example problem} \\ &\dfrac{10t}{6t − 12} ∗ \dfrac{20(t − 2)}{30t^3} &\text{Factor all numerators and denominators} \\ &\dfrac{(2)(5)(t)(2)(2)(5)(t − 2)}{(2)(3)(t − 2)(2)(3)(5)(t)(t^2)} &\text{Factor numbers into prime factors and write with one division bar} \\ &\dfrac{\cancel{(2)}(5)\cancel{(t)}(2)(2)\cancel{(5)}\cancel{(t − 2)}}{\cancel{(2)}(3)\cancel{(t − 2)}(2)(3)\cancel{(5)}\cancel{(t)}(t^2)} &\text{Remove common factors} \\ &\dfrac{(5)(2)(2)}{(3)(2)(3)(t^2)} &\text{Simplify} \\ &\dfrac{20}{18t^2} &\text{Final answer (it is good practice to show final answer as factored as possible)} \end{array} \)
    1. \(\begin{array} &&\dfrac{7x + 14}{2x^2 − 8} ∗ \dfrac{x^2 + 3x − 10}{14x + 21} &\text{Example problem} \\ &\dfrac{7(x + 2)}{2(x^2 − 4)} ∗ \dfrac{(x + 5)(x − 2)}{7(2x + 3)} &\text{Factor all numerators and denominators} \\ &\dfrac{7(x + 2)(x + 5)(x − 2)}{2(x + 2)(x − 2)(7)(2x + 3)} &\text{Further factor algebraic expressions and numbers into prime factors and write with one division bar} \\ &\dfrac{\cancel{7}\cancel{(x + 2)}(x + 5)(x − 2)}{2\cancel{(x + 2)}(x − 2)\cancel{(7)}(2x + 3)} &\text{Remove common factors} \\ &\dfrac{(x + 5)(x − 2)}{2(x − 2)(2x + 3)} &\text{Final answer} \end{array}\)
    1. \(\begin{array} &&\dfrac{3x^3 − 24}{2x^2 − 14x + 20} ∗ \dfrac{4x^3 − 20x^2 + 3x − 15}{3x^2 + 6x + 12} &\text{Example problem} \\ &\dfrac{3(x^3 − 8)}{2(x^2 − 7x + 10)} ∗ \dfrac{4x^2 (x − 5) + 3(x − 5)}{3(x^2 + 2x + 4)} &\text{Factor all numerators and denominators. Use factor by grouping for the \(4\)-term 多项式。}\\ &\ dfrac {3 (x − 2) (x^2 + 2x + 4)} {2 (x − 5) (x − 2)} ∗\ dfrac {(4x^2 + 3) (x − 5)} {3 (x + 2)} &\ text {进一步因子代数表达式}\\ &\ dfrac {3 (x − 2) (x^2 + 2x + 4) (4x^2 + 3) (x − 5)} {2 (x − 5) (x − 2) (3) (x + 2)} &\ text {进一步因子代数表达式和数字转换为素数然后用一个分条写入}\\ &\ dfrac {\ cancel {3}\ cancel {(x − 2)} (x^2 + 4) (4x^2 + 3)\ cancel {(x − 5)}\ cancel {(x − 2)}\ cancel {(x − 2)} (x 2) (x 2) (x + 2)} &\ text {移除常见因素}\\ &\ dfrac {(x^2 + 2x + 4) (4x^2 + 3)} {2 (x + 2) (x + 2)} &\ text {最终答案}\\ &\ dfrac {(x^2 + 2x + 4) (4x^2 + 3)} {2 (x + 2) ^2} &\ text {最终答案也可以用这种形式写}\ end {array}\)
    练习 Template:Index

    将有理表达式相乘：

    1. \(\dfrac{x^2 + 4x + 3}{2x^2 − x − 10} ∗ \dfrac{2x^2 + 4x^3}{x^2 + 3x} ∗ \dfrac{x}{x^2 + 3x + 2}\)
    2. \(\dfrac{x^2 + 2x − 15}{x^2 − 4x − 45} ∗ \dfrac{x^2 − 5x − 36}{x^2 + x − 12}\)
    3. \(\dfrac{x^2 + 3x − 40}{x^2 + 2x − 35} ∗ \dfrac{x^2 + 3x − 18}{4x^2 − 5x − 32x + 40}\)