课程： 9.9 分裂的理性表达式

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选择节Divide 理性表达式

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Divide 理性表达式
The area of a rectangle is $\begin{align*}\frac{12x^2yz^3}{5xy^2z}\end{align*}$ . The length of the rectangle is $\begin{align*}\frac{2xy}{z^2}\end{align*}$ . What is the width of the rectangle?
::矩形区域为 12x2yz35xy2z。矩形的长度为 2xyz2。矩形的宽度是多少?

Dividing Rational Expressions
::Divide 理性表达式

Dividing rational expressions requires one more step than multiplying them does. Recall that when you divide fractions, you need to flip the second fraction and change the problem to multiplication . The same rule applies to dividing rational expressions.
::偏移理性表达比乘以它们需要多一步。提醒您注意, 当分割分数时, 您需要翻转第二个分数, 并将问题改为乘法。同一规则适用于分解理性表达。

D ivide the following rational expressions.
::将下列合理表达式分开。

$\begin{align*}\frac{5a^3b^4}{12ab^8} \div \frac{15b^6}{8a^6}\end{align*}$
::5a3b412ab815b68a6

Flip the second fraction, change the $\begin{align*}\div\end{align*}$ sign to multiplication, and simplify .
::翻转第二个分数, 将 {} 符号更改为乘法, 并简化。

$\begin{align*}\frac{5a^3b^4}{12ab^8} \div \frac{15b^6}{8a^6} = \frac{5a^3b^4}{12ab^8} \cdot \frac{8a^6}{15b^6} = \frac{40a^9b^4}{180ab^{14}} = \frac{2a^8}{9b^{10}}\end{align*}$
::5a3b412AB815b68a6=5a3b412ab8_8a615b6=40a9b4180AB14=2a89b10

$\begin{align*}\frac{x^4-3x^2-4}{2x^2+x-10} \div \frac{x^3-3x^2+x-3}{x-2}\end{align*}$
::x4 - 3x2 - 42x2+x- 103 - 3x2+x-3x-2

Flip the second fraction, change the $\begin{align*}\div\end{align*}$ sign to multiplication and simplify .
::翻转第二个分数,将 {} 符号更改为乘法和简化。

$\begin{aligned} \frac{x^{4} - 3 x^{2} - 4}{2 x^{2} + x - 10} \div \frac{x^{3} - 3 x^{2} + x - 3}{x - 2} & = \frac{x^{4} - 3 x^{2} - 4}{2 x^{2} + x - 10} \cdot \frac{x - 2}{x^{3} - 3 x^{2} + x - 3} \\ = \frac{(x^{2} - 4) (x^{2} + 1)}{(2 x - 5) (x + 2)} \cdot \frac{x - 2}{(x^{2} + 1) (x - 3)} \\ = \frac{(x - 2) (x + 2) (x^{2} + 1)}{(2 x - 5) (x + 2)} \cdot \frac{x - 2}{(x^{2} + 1) (x - 3)} \\ = \frac{(x - 2)^{2}}{(2 x - 5) (x - 3)} \end{aligned}$
$\begin{align*}\frac{x^4-3x^2-4}{2x^2+x-10} \div \frac{x^3-3x^2+x-3}{x-2} &= \frac{{\color{red}x^4-3x^2-4}}{2x^2+x-10} \cdot \frac{x-2}{{\color{blue}x^3-3x^2+x-3}} \\ &= \frac{(x^2-4)(x^2+1)}{(2x-5)(x+2)} \cdot \frac{x-2}{(x^2+1)(x-3)} \\ &= \frac{(x-2)\cancel{(x+2)} \cancel{(x^2+1)}}{(2x-5)\cancel{(x+2)}} \cdot \frac{x-2}{\cancel{(x^2+1)}(x-3)} \\ &= \frac{(x-2)^2}{(2x-5)(x-3)}\end{align*}$
::x4 - 3x2 - 3x2 - 3x2 - 4x2 - 4x2 3x2 - 3x2x2 - 4x2 - 3x2 - 4x2 - 3x2 - 3x2 - 3x2 - 3x2 - 3x2 - 3x2 - 3x2 - 3x2 - 3x2 - 42x2 - 3x2 - 3x2 - 3x2 - 3x3= (x2 - 4x2 - 4x2x2 - 4x2x2 - 4x2x2 - 4x2+1(x1)(2x-5) = (x-2 - 3)

Now perform the indicated operations : $\begin{align*}\frac{{\color{blue}x^3-8}}{x^2-6x+9} \div (x^2+3x-10) \cdot \frac{x^2+x-12}{x^2+11x+30}\end{align*}$ .
::现在执行所示操作: x3-8x2-6x+9(x2+3x-10)x2+x-12x2+11x+30。

Flip the second term , factor , and cancel (remember $\begin{align*}x^3 -8\end{align*}$ is a difference of cubes).
::翻转第二个任期、系数和取消(记住 x3-8 是立方体的差数) 。

$\begin{aligned} \frac{x^{3} - 8}{x^{2} - 6 x + 9} \div (x^{2} + 3 x - 10) \cdot \frac{x^{2} + x - 12}{x^{2} + 11 x + 30} & = \frac{x^{3} - 8}{x^{2} - 6 x + 9} \cdot \frac{1}{x^{2} + 3 x - 10} \cdot \frac{x^{2} + 2 x - 15}{x^{2} + 11 x + 30} \\ = \frac{(x - 2) (x^{2} + 2 x + 4)}{(x - 3) (x - 3)} \cdot \frac{1}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 3)}{(x + 5) (x + 6)} \\ = \frac{x^{2} + 2 x + 4}{(x - 3) (x + 5) (x + 6)} \end{aligned}$
$\begin{align*}\frac{x^3-8}{x^2-6x+9} \div (x^2+3x-10) \cdot \frac{x^2+x-12}{x^2+11x+30} &= \frac{x^3-8}{x^2-6x+9} \cdot \frac{1}{x^2+3x-10} \cdot \frac{x^2+2x-15}{x^2+11x+30} \\ &= \frac{\cancel{(x-2)}(x^2+2x+4)}{\cancel{(x-3)}(x-3)} \cdot \frac{1}{\cancel{(x-2)} \cancel{(x+5)}} \cdot \frac{\cancel{(x+5)} \cancel{(x-3)}}{(x+5)(x+6)} \\ &= \frac{x^2+2x+4}{(x-3)(x+5)(x+6)}\end{align*}$
::x3 - 8x2 - 6x+9 (x2+3x- 10) x2+xx12x2+11x30=x12x2+30=x3x3-8x2 - 6x6x+9_1x2+3x- 10xxxx2+2x- 15x2+2x- 15x2+15x2+11xx=(x-2)(x2+2+2xx4+4)(x3)(x-3)(x3)(x-3)(x3)(x3)(x3)(x3)(x+5)(x+6) =x2+2x4(x-3)(x+5)(x+6)

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the width of a rectangle.
::早些时候,有人要求你找到一个矩形的宽度。

To find the width, divide the area by the length and simplify.
::要找到宽度, 请将区域除以长度并简化。

$\begin{aligned} \frac{12 x^{2} y z^{3}}{5 x y^{2} z} \div \frac{2 x y}{z^{2}} \\ \frac{12 x^{2} y z^{3}}{5 x y^{2} z} \cdot \frac{z^{2}}{2 x y} \\ \frac{12 x^{2} y z^{5}}{10 x^{2} y^{3} z} \\ \frac{6 z^{4}}{5 y^{2}} \end{aligned}$
$\begin{align*}\frac{12x^2yz^3}{5xy^2z} \div \frac{2xy}{z^2}\\ \frac{12x^2yz^3}{5xy^2z} \cdot \frac{z^2}{2xy}\\ \frac{12x^2yz^5}{10x^2y^3z}\\ \frac{6z^4}{5y^2}\end{align*}$
::12x2yz35xy2z2xyz212x2yz35xy2z22xy12x5z510x2y3z6z45y2

Therefore , the width of the rectangle is $\begin{align*}\frac{6z^4}{5y^2}\end{align*}$ .
::因此,矩形宽度为6z45y2。

Perform the indicated operations.
::执行指定的操作。

Example 2
::例2

$\begin{align*}\frac{a^5b^3c}{6a^2c^9} \div \frac{2a^7b^{11}}{24c^2}\end{align*}$
::5b3c6a2c92a7b1124c2

Invert the second fraction and simplify:
::倒转第二个分数并简化 :

$\begin{align*}\frac{a^5b^3c}{6a^2c^9} \div \frac{2a^7b^{11}}{24c^2} = \frac{a^5b^3c}{6a^2c^9} \cdot \frac{24c^2}{2a^7b^{11}} = \frac{24a^5b^3c^3}{12a^9b^{11}c^9} = \frac{2}{a^4b^8c^6}\end{align*}$
::a5b3c6a2c92a7b1124c2=a5b3c6a2c924c22a7b11=24a5b3c312a9b11c9=2a4b8c6

Example 3
::例3

$\begin{align*}\frac{x^2+12x-45}{x^2-5x+6} \div \frac{x^2+17x+30}{x^4-16}\end{align*}$
::x2+12x-45x2-5x+6x2+17x+30x4-16

Invert the second fraction and simplify:

$\begin{aligned} \frac{x^{2} + 12 x - 45}{x^{2} - 5 x + 6} \div \frac{x^{2} + 17 x + 30}{x^{4} - 16} & = \frac{x^{2} + 12 x - 45}{x^{2} - 5 x + 6} \cdot \frac{x^{4} - 16}{x^{2} + 17 x + 30} \\ = \frac{(x + 15) (x - 3)}{(x - 3) (x - 2)} \cdot \frac{(x^{2} + 4) (x - 2) (x + 2)}{(x + 15) (x + 2)} \\ = x^{2} + 4 \end{aligned}$
$\begin{align*}\frac{x^2+12x-45}{x^2-5x+6} \div \frac{x^2+17x+30}{x^4-16} &= \frac{x^2+12x-45}{x^2-5x+6} \cdot \frac{x^4-16}{x^2+17x+30} \\ &= \frac{\cancel{(x+15)}\cancel{(x-3)}}{\cancel{(x-3)} \cancel{(x-2)}} \cdot \frac{(x^2+4)\cancel{(x-2)} \cancel{(x+2)}}{\cancel{(x+15)} \cancel{(x+2)}} \\ &= x^2+4\end{align*}$
::倒转第二个分数和简化 : x2+12x- 45x2- 5x=x2+12x- 30x4- 16=x2+12x- 30x4=x12x- 45x2- 5x+6x+6x4- 16x2+17x+30=(x+15)(x-3(x-3)(x-3)(x-3)(x-2)(x-2)(x-2)(x-2)(x-2)(x+2)(x+2)(x+2)(x+15)(x+2)=x2+4

Example 4
::例4

$\begin{align*}(x^3+2x^2-9x-18) \div \frac{x^2+11x+24}{x^2-11x-24} \div \frac{x^2-6x-16}{x^2+5x-24}\end{align*}$
:x3+2x2-9x-2-9x-1818)x2+11x+24x2-11x-24x2-6x-16x2+5x-24)

Write the first term over one, invert the second and third fractions, and simplify:

$\begin{aligned} (x^{3} + 2 x^{2} - 9 x - 18) \div \frac{x^{2} + 11 x + 24}{x^{2} - 11 x + 24} \div \frac{x^{2} - 6 x - 16}{x^{2} + 5 x - 24} & = \frac{x^{3} + 2 x^{2} - 9 x - 18}{1} \cdot \frac{x^{2} - 11 x + 24}{x^{2} + 11 x + 24} \cdot \frac{x^{2} + 5 x - 24}{x^{2} - 6 x - 16} \\ = \frac{(x - 3) (x + 3) (x + 2)}{1} \cdot \frac{(x - 8) (x - 3)}{(x + 8) (x + 3)} \cdot \frac{(x + 8) (x - 3)}{(x - 8) (x + 2)} \\ = (x - 3)^{2} \end{aligned}$
$\begin{align*}(x^3+2x^2-9x-18) \div \frac{x^2+11x+24}{x^2-11x+24} \div \frac{x^2-6x-16}{x^2+5x-24} &= \frac{x^3+2x^2-9x-18}{1} \cdot \frac{x^2-11x+24}{x^2+11x+24} \cdot \frac{x^2+5x-24}{x^2-6x-16} \\ &= \frac{(x-3) \cancel{(x+3)} \cancel{(x+2)}}{1} \cdot \frac{\cancel{(x-8)}(x-3)}{\cancel{(x+8)}\cancel{(x+3)}} \cdot \frac{\cancel{(x+8)}(x-3)}{\cancel{(x-8)} \cancel{(x+2)}} \\ &=(x-3)^2\end{align*}$
::将第二和第三个分数倒转, 并简化 : (x3+2x2- 9x2- 9x-18) *x2+11x+24x2- 11x2- 11xx+24x*x2- 6x- 16x2+5x2- 6x2=x3+2x2- 9x2- 9x2- 181x2x2- 11x+24x2+11x2+24x2+24x2+24x2+5x24x2- 6x16=(x-3x+3)(x+2)1x(8)(x-8)(x+3)-(x+8)(x3)(x- 8)(x-8)(x)(x-8)(x-8)(x)(x-8)(x+2)=(x- 3)

Review
::回顾

Divide the following expressions. Simplify your answer.
::除以下列表达式。简化您的回答。

$\begin{align*}\frac{6a^4b^3}{8a^3b^6} \div \frac{3a^5}{4a^3b^4}\end{align*}$
::6a4b38a3b6*3a54a3b4

$\begin{align*}\frac{12x^5y}{xy^4} \div \frac{18x^3y^6}{3x^2y^3}\end{align*}$
::12x5yxy418x363x2y3

$\begin{align*}\frac{16x^3y^9z^3}{15x^5y^2z} \div \frac{42xy^7z^2}{45x^2yz^5}\end{align*}$
::16x3y9z315x5y2z42xy7z245x2yz5

$\begin{align*}\frac{x^2+2x-3}{x^2-3x+2} \div \frac{x^2+3x}{4x-8}\end{align*}$
::x2+2x-3x2 -3x2 -3x2\%x2+3x4x-8

$\begin{align*}\frac{x^2-2x-3}{x^2+6x+5} \div \frac{4x-12}{x^2+8x+15}\end{align*}$
::x2 - 2x - 3x2+6x+5**4x - 12x2+8x+15

$\begin{align*}\frac{x^2+6x+2}{12-3x} \div \frac{6x^2-13x-5}{x^2-4x}\end{align*}$
::x2+6x+212-3x=6x2-13x-5x2-4x

$\begin{align*}\frac{x^2-5x}{x^2+x-6} \div \frac{x^2-2x-15}{x^3+3x^2-4x-12}\end{align*}$
::x2 - 5xx2+x - 6x2 - 2x - 15x3+3x2 - 4x- 12

$\begin{align*}\frac{3x^3-3x^2-6x}{2x^2+15x-8} \div \frac{6x^2+18x-60}{2x^2+9x-5}\end{align*}$
::3x3 - 3x2 - 6x2x2+15x-86x2+18x- 602x2+9x-5

$\begin{align*}\frac{x^3+27}{x^2+5x-14} \div \frac{x^2-x-12}{2x^2+2x-40} \div \frac{1}{x-2}\end{align*}$
::x3+27x2+5x-14_x2-x-122x2+2x-40_1x-2

$\begin{align*}\frac{x^2+2x-15}{2x^3+7x^2-4x} \div (5x+3) \div \frac{21-10x+x^2}{5x^3+23x^2+12x}\end{align*}$
::x2+2x - 152x3+7x2_ 4x( 5x+3) @%21- 10x+x25x3+23x2+12x

We all know that when you divide fractions, you take the second fraction, flip it, and change it to a multiplication problem. But, do you know why? Let's investigate the why here.
::我们都知道,当你分裂分数时, 你拿第二分数, 翻转它, 把它变成乘法问题。但是, 你知道为什么吗? 我们来调查一下为什么在这里。

What is $\begin{align*}6 \div 2\end{align*}$ ?
::什么是62?

What about $\begin{align*}\frac{1 \div 1}{6 \div 2}\end{align*}$ ?
::怎么样1162?

Is the problem above the same as $\begin{align*}\frac{1}{6} \div \frac{1}{2}\end{align*}$ ? Why or why not?
::问题是否与16+12相同?为什么或为什么没有?

Let's take a different approach. Let's write a division problem as a huge fraction: $\begin{align*}\frac{\frac{30}{52}}{\frac{15}{13}}\end{align*}$
::让我们采取不同的方法。让我们把分裂问题写成一个巨大的部分:30521513。

We know we cannot have fractions in the denominator of another fraction. What would we have to multiply the denominator by to cancel it out?
::我们知道,我们无法在分母的分母中找到分数。我们有什么用乘法来取消分母呢?

Multiply the top and bottom from your answer in #14. What did you multiply by?
::乘以在 #14 中的答案的顶部和底部。您乘以了什么 ?

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。

章节大纲

常规

Divide 理性表达式

Dividing Rational Expressions ::Divide 理性表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Dividing Rational Expressions
::Divide 理性表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)