Course: 12.9 理性表达方式司

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理性表达式分区
Division of Rational Expressions
::理性表达式分区

Just as with ordinary fractions, we first rewrite the division problem as a multiplication problem and then proceed with the multiplication as outlined in the previous section.
::如同普通分数一样,我们首先将分数问题改写为乘法问题,然后按前一节所述进行乘法。

Note: Remember that $\begin{aligned} \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} \end{aligned}$ . The first fraction remains the same and you take the reciprocal of the second fraction. Do not fall into the common trap of flipping the first fraction.
::注意: 记住 abcd=abdc 。第一个分数保持不变, 您可以对等第二个分数。不要掉入翻转第一个分数的常见陷阱。

Divide $\begin{aligned} \frac{4 x^{2}}{15} \div \frac{6 x}{5} \end{aligned}$ .
::除以 4x215\\\\\6x5。

First convert into a multiplication problem by flipping the second fraction and then simplify as usual:
::首先翻转第二个分数, 转换成乘法问题, 然后像往常一样简化 :

$\begin{aligned} \frac{4 x^{2}}{15} \div \frac{6 x}{5} = \frac{4 x^{2}}{15} \cdot \frac{5}{6 x} = \frac{2 x}{3} \cdot \frac{1}{3} = \frac{2 x}{9} \end{aligned}$

::4x215=6x5=4x215*56x=2x3}13=2x9

Dividing a Rational Expression by a Polynomial
::以多面性表示逻辑表达式

When we divide a rational expression by a whole number or a polynomial, we can write the whole number (or polynomial) as a fraction with denominator equal to one, and then proceed the same way as in the previous examples.
::当我们用一个整数或多数值来分隔一个理性表达式时, 我们可以将整个数字( 或多数值) 写成一个分数, 分母等于一个分母, 然后以与前几个例子相同的方式进行。

Divide $\begin{aligned} \frac{9 x^{2} - 4}{2 x - 2} \div (21 x^{2} - 2 x - 8) \end{aligned}$ .
::除号 9x2 - 42x - 2 (21x2 - 2x-8)。

Rewrite the expression as a division of fractions, and then convert into a multiplication problem by taking the reciprocal of the divisor:
::将表达式重写为分数的分割, 然后通过对等的断点转换成乘法问题 :

$\begin{aligned} \frac{9 x^{2} - 4}{2 x - 2} \div \frac{21 x^{2} - 2 x - 8}{1} = \frac{9 x^{2} - 4}{2 x - 2} \cdot \frac{1}{21 x^{2} - 2 x - 8} \end{aligned}$

::9x2 - 42x- 2221x2 - 2x- 81=9x2 - 42x-2 - 22=21x2 - 2x2 - 2x8

Then factor and solve:
::然后点和解答 :

$\begin{aligned} \frac{9 x^{2} - 4}{2 x - 2} \cdot \frac{1}{21 x^{2} - 2 x - 8} = \frac{(3 x - 2) (3 x + 2)}{2 (x - 1)} \cdot \frac{1}{(3 x - 2) (7 x + 4)} = \frac{(3 x + 2)}{2 (x - 1)} \cdot \frac{1}{(7 x + 4)} = \frac{3 x + 2}{14 x^{2} - 6 x - 8} \end{aligned}$

::9x2 - 42x - 218-121x2 - 2x-8=(3x-2)(3x-2)(3x+2)(2)(x)-11(3x-2)(7x+4)=(3x+2)(2)(2)(2)(x-1)(7x+4)=(3x+2)(2)(2)(2)(2)(7x+2)(7x+4)(7x+4)=3x+214x2-6x-8)

Solve Applications Involving Multiplication and Division of Rational Expressions
::涉及乘数和合理表达式分工的解决应用程序

Suppose Marciel is training for a running race. Marciel’s speed (in miles per hour) of his training run each morning is given by the function $\begin{aligned} x^{3} - 9 x \end{aligned}$ , where $\begin{aligned} x \end{aligned}$ is the number of bowls of cereal he had for breakfast. Marciel’s training distance (in miles), if he eats $\begin{aligned} x \end{aligned}$ bowls of cereal, is $\begin{aligned} 3 x^{2} - 9 x \end{aligned}$ . What is the function for Marciel’s time, and how long does it take Marciel to do his training run if he eats five bowls of cereal on Tuesday morning?
::假设Marciel是赛跑的训练对象。 Marciel每天早上的训练速度(每小时每英里)是由函数x3-9x提供的,这里x是他早餐时的谷物碗数。 Marciel的训练距离(每英里)是3x2-9x。 Marciel如果在星期二早上吃五碗谷物,那么他要多久才能参加训练?

$\begin{aligned} time = \frac{distance}{speed} \\ time = \frac{3 x^{2} - 9 x}{x^{3} - 9 x} = \frac{3 x (x - 3)}{x (x^{2} - 9)} = \frac{3 x (x - 3)}{x (x + 3) (x - 3)} \\ time = \frac{3}{x + 3} \\ If x = 5, then \\ time = \frac{3}{5 + 3} = \frac{3}{8} \end{aligned}$

::时间= 距离时= 3x2- 9x3- 9x3x3x3x3xxx2- 9x3x3xxx3xx3xxx3x3x3x3x3x3x3x3x3xx3x3Ifx=5,然后时间=35+3=38

Marciel will run for $\begin{aligned} \frac{3}{8} \end{aligned}$ of an hour.
::Marciel将跑38小时

Example
::示例示例示例示例

Example 1
::例1

Divide $\begin{aligned} \frac{3 x^{2} - 15 x}{2 x^{2} + 3 x - 14} \div \frac{x^{2} - 25}{2 x^{2} + 13 x + 21} \end{aligned}$ .
::除以 3x2 - 15x2x2+3x- 14}x2 - 252x2+13x+21。

$\begin{aligned} \frac{3 x^{2} - 15 x}{2 x^{2} + 3 x - 14} \cdot \frac{2 x^{2} + 13 x + 21}{x^{2} - 25} = \frac{3 x (x - 5)}{(2 x + 7) (x - 2)} \cdot \frac{(2 x + 7) (x + 3)}{(x - 5) (x + 5)} = \frac{3 x}{(x - 2)} \cdot \frac{(x + 3)}{(x + 5)} = \frac{3 x^{2} + 9 x}{x^{2} + 3 x - 10} \end{aligned}$

::3x2 - 15x2x2x2+3x3x- 14=2x2x2x2+13x+21x2x2-2-225=3xx(x-5)(2x-5)(2x+7)(x-7)(x-2)=(2x+7)(x+3)(x-5)(x+5)(5x+5)=3x(x-2)(x+5)(x+5)(x+3+3(3+3(3)(x+3)(x+5)=3x2+9xx2+3x-10)

Review
::回顾

Divide the rational functions and reduce the answer to lowest terms.
::将合理功能分开,将答案降低到最低水平。

$\begin{aligned} 2 x y \div \frac{2 x^{2}}{y} \end{aligned}$
::2x%2x2y

$\begin{aligned} \frac{2 x^{3}}{y} \div 3 x^{2} \end{aligned}$
::2x3y3x2

$\begin{aligned} \frac{3 x + 6}{y - 4} \div \frac{3 y + 9}{x - 1} \end{aligned}$
::3x+6y- 43y+9x-1

$\begin{aligned} \frac{x^{2}}{x - 1} \div \frac{x}{x^{2} + x - 2} \end{aligned}$
::x2x- 1%x2+x-2

$\begin{aligned} \frac{a^{2} + 2 a b + b^{2}}{a b^{2} - a^{2} b} \div (a + b) \end{aligned}$
::a2+2ab+b2ab2-a2b*(a+b)

$\begin{aligned} \frac{3 - x}{3 x - 5} \div \frac{x^{2} - 9}{2 x^{2} - 8 x - 10} \end{aligned}$
::3-x3x-5x2-92x2-8x-10

$\begin{aligned} \frac{x^{2} - 25}{x + 3} \div (x - 5) \end{aligned}$
::x2 - 25x+3(x-5)

$\begin{aligned} \frac{2 x + 1}{2 x - 1} \div \frac{4 x^{2} - 1}{1 - 2 x} \end{aligned}$
::2x+12x-14x2-11-2x

$\begin{aligned} \frac{3 x^{2} + 5 x - 12}{x^{2} - 9} \div \frac{3 x - 4}{3 x + 4} \end{aligned}$
::3x2+5x-12x2-93x-43x+4

$\begin{aligned} \frac{x^{2} + x - 12}{x^{2} + 4 x + 4} \div \frac{x - 3}{x + 2} \end{aligned}$
::x2+x- 12x2+4x+4x-3x+2

$\begin{aligned} \frac{x^{4} - 16}{x^{2} - 9} \div \frac{x^{2} + 4}{x^{2} + 6 x + 9} \end{aligned}$
::x4 - 16x2 - 92+4x2+6x+9

Maria’s recipe asks for $\begin{aligned} 2 \frac{1}{2} \end{aligned}$ times as much flour as sugar. How many cups of flour should she mix in if she uses $\begin{aligned} 3 \frac{1}{3} \end{aligned}$ cups of sugar?
::玛丽亚的食谱要求的面粉数量是糖的212倍。如果她使用313杯糖,她应该混合多少杯面粉?

George drives from San Diego to Los Angeles. On the return trip he increases his driving speed by 15 miles per hour. In terms of his initial speed, by what factor is the driving time decreased on the return trip?
::George开车从圣地亚哥到洛杉矶。在回程中,他将车速提高15英里/小时。从最初的速度看,回程的驾驶时间因何而缩短?

Ohm’s Law states that in an electrical circuit $\begin{aligned} I = \frac{V}{R_{t o t}} . \end{aligned}$ The total resistance for resistors placed in parallel is given by: $\begin{aligned} \frac{1}{R_{t o t}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} . \end{aligned}$ Write the formula for the electric current in terms of the component resistances: $\begin{aligned} R_{1} \end{aligned}$ and $\begin{aligned} R_{2} . \end{aligned}$
::Ohm的法律规定,在电路I=VRtot中,对平行置放的阻力的完全抵抗力是由:1Rtot=1R1+1R2提供的。用部件阻力来写电流的公式:1R1和R2。

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

理性表达式分区

Division of Rational Expressions ::理性表达式分区

Solve Applications Involving Multiplication and Division of Rational Expressions ::涉及乘数和合理表达式分工的解决应用程序

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Division of Rational Expressions
::理性表达式分区

Solve Applications Involving Multiplication and Division of Rational Expressions
::涉及乘数和合理表达式分工的解决应用程序

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)