节: 理性表达式 | 2.4 理性表达式

章节大纲

A rational expression is a ratio, just like a fraction . However, instead of a ratio between numbers, a rational expression is a ratio between two expressions.

One driving question to ask is: Are the rules for simplifying and operating on rational expressions the same as the rules for simplifying and operating on fractions?

Working with Rational Expressions
::与有理表达式一起工作

When simplifying or operating on rational expressions, it is vital that each polynomial be fully factored. Once all expressions are factored, identical factors in the numerator and denominator may be canceled or removed. The reason they can be removed is that any expression divided by itself is equal to 1. An identical expression in the numerator and denominator is just an expression being divided by itself, and so equals 1.
::当在理性表达式上简化或操作时,每个多式表达式都必须被充分计算。一旦所有表达式都计算在内, 分子和分母中的相同因素可以取消或删除。它们可以删除的原因是, 任何自除的表达式都等于 1. 分子和分母中的相同表达式只是一个本身被分割的表达式, 因而等于 1 。

Adding and Subtracting Rational Expressions
::添加和减减理性表达式

To add or subtract rational expressions, it is essential to first find a common denominator. While any common denominator will work, using the least common denominator is a means of keeping the number of additional factors under control. Look at each rational expression you are working with and identify your desired common denominator. Multiply each expression by an appropriate form of 1, such as $\begin{aligned} \frac{x - 2}{x - 2} \end{aligned}$ , and then you should have your common denominator. In addition and subtraction problems, the numerator must be multiplied, combined, and re-factored to be considered simplified.
::要添加或减去理性表达式,首先必须找到一个共同的分母。任何共同的分母都会有效, 使用最小的共同分母是控制额外因素数量的一种手段。查看您正在使用的每种合理表达式, 并确定您想要的共分母。将每个表达式乘以适当的 1 形式, 如 x- 2x-2, 然后您应该拥有共同的分母。此外, 减法问题 , 分子必须乘以、合并和重新设定, 才能被视为简化。

Multiplying and Dividing Rational Expressions
::乘法和分裂逻辑表达式

To multiply rational expressions, you should write the product of all the numerator factors over the product of all the denominator factors and then cancel, or remove, identical factors.To divide rational expressions, you should rewrite the division problem as a multiplication problem. Multiply the first rational expression by the reciprocal of the second rational expression. Follow the steps above for multiplying. In both multiplication and division problems answers are most commonly left entirely factored to demonstrate everything has been reduced appropriately.
::要乘以理性表达式, 您应该在所有分母要素的产物上写入所有分子系数的产物, 然后取消或删除相同因素。要将合理表达式分开, 您应该将分裂问题重写为乘法问题。乘以第一个合理表达式, 乘以第二个合理表达式的对等。顺着以上步骤进行乘法。在乘法和分法问题回答中, 通常都留有全部因素来显示所有问题都得到了适当的减少。

Examples
::实例

Example 1
::例1

Earlier, you were asked if the rules for simplifying and operating on rational expressions are the same as the rules for simplifying and operating on fractions. R ational expressions are an extension of fractions and the operations of simplifying, adding, subtracting, multiplying and dividing work in exactly the same way.
::早些时候,有人问您,简化和操作理性表达的规则是否与简化和操作分数的规则相同。理性表达是分数的延伸,是简化、增减、增减、倍增和拆分工作的操作,其方式完全相同。

Example 2
::例2

Subtract the following rational expressions.
::减去以下合理表达式。

\begin{aligned} \frac{x - 2}{x + 3} - \frac{x^{3} - 3 x^{2} + 8 x - 24}{2 (x + 2) (x^{2} - 9)} \end{aligned}

::x-2x+3-3-x3-3-3x2+8x-242(x+2)(x2-9)

Being able to factor effectively is of paramount importance.
::能够有效地考虑因素至关重要。

\begin{aligned} = \frac{x - 2}{x + 3} - \frac{x^{3} - 3 x^{2} + 8 x - 24}{2 (x + 2) (x^{2} - 9)} \\ = \frac{(x - 2)}{(x + 3)} - \frac{x^{2} (x - 3) + 8 (x - 3)}{2 (x + 2) (x^{2} - 9)} \\ = \frac{(x - 2)}{(x + 3)} - \frac{(x - 3) (x^{2} + 8)}{2 (x + 2) (x + 3) (x - 3)} \end{aligned}

::=x-2x+3-x3-x3-x3-x3-x3-x3-x3-x2-x8-2-8-2-2-x2(x2+2)(x2-9)=(x-2(x+3)-x2(x2-3)-(x-2(x2-2)-9)=(x-2(x-2)(x2-2)-(x3)-(x-3)(x-3)(x2+8)-2(x+2)(x+3)(x-3)

Before subtracting, simplify where possible so you don’t contribute to unnecessarily complicated denominators.
::在减去之前,尽可能简化,这样你就不会造成不必要的复杂分母。

\begin{aligned} = \frac{(x - 2)}{(x + 3)} - \frac{x^{2} + 8}{2 (x + 2) (x + 3)} \end{aligned}

::=(x--2)(x+3)-x2+82(x+2)(x+3)

The left expression lacks $\begin{aligned} 2 (x + 2) \end{aligned}$ , so multiply both its numerator and denominator by $\begin{aligned} 2 (x + 2) \end{aligned}$ .
::左方表达式缺少2(x+2), 所以将其分子和分母乘以2( x+2) 。

$\begin{aligned} = \frac{2 (x + 2) (x - 2)}{2 (x + 2) (x + 3)} - \frac{(x^{2} + 8)}{2 (x + 2) (x + 3)} \\ = \frac{2 (x^{2} - 4) - x^{2} - 8}{2 (x + 2) (x + 3)} \\ = \frac{x^{2} - 16}{2 (x + 2) (x + 3)} \end{aligned}$
Example 3
::=2x+2(x+2)(x-2)(x+2)(x+3)-(x2+8)(x2)(x+3)=2x2-4)(x2-82(x+2)(x+3)(x+3)=x2-162(x+2)(x+3)例3

Simplify the following rational expression.
::简化以下合理表达式。

\begin{aligned} \frac{x^{2} + 7 x + 12}{x^{2} + 4 x + 3} \cdot \frac{x^{2} + 9 x + 8}{2 x^{2} - 128} \div \frac{x + 4}{x - 8} \cdot \frac{14}{1} \end{aligned}

::x2+7x+12x2+4x+3xx2+9x+82x2_128_x+4x_8}141

First factor everything. Second, turn division into multiplication (only one term). Third, cancel appropriately which will leave the answer.
::第一个因素。第二,将分数转换成乘法( 只有一个条件 ) 。第三, 适当取消, 这样会留下答案。

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

::==(x+3)(x+3)(x+3)(x+3)(x+3)(x+3)(x+1)(x1)(x+8)(x+8)(x-8)(x-8)(x-8)(x-8)(x-+4)(4)(141)=(x+3)(x+4)(x+3)(x+3(x+3)(x+1)(x+8)(x+1)(x+8)(x-8)(x-8)(x-8)(x-8)(x-8)(x-8)(x+4)(141)=142=7)

In this example, the strike through is shown. You should use this technique to match up factors in the numerator and the denominator when simplifying.
::在此示例中显示罢工过程。您应该使用这种技术来匹配分子中的系数和简化时的分母。

Example 4
::例4

Combine the following rational expressions.
::将下列合理表达式合并。

\begin{aligned} \frac{1}{x^{2} + 5 x + 6} - \frac{1}{x^{2} - 4} + \frac{(x - 7) (x + 5) + 5}{(x + 2) (x - 2) (x + 3) (x - 4)} \end{aligned}

::1x2+5x+6-1x2-4+(x-7)(x+5)+5(x+2)(x-2)(x+3)(x-4)

First factor everything and decide on a common denominator. While the numerators do not really need to be factored, it is sometimes helpful in simplifying individual expressions before combining them. Note that the numerator of the expression on the right hand seems factored but it really is not. Since the 5 is not connected to the rest of the numerator through multiplication, that part of the expression needs to be multiplied out and like terms need to be combined.
::第一个因素是所有事物,然后决定一个共同的分母。虽然点数并不真的需要计算, 但有时会有助于简化单个表达式, 然后将其合并。请注意, 右手表达式的点数似乎已经计算, 但实际上并没有计算。由于 5 没有通过乘法连接到分子的其余部分, 这部分表达式需要乘出, 类似术语需要合并。

\begin{aligned} = \frac{1}{(x + 2) (x + 3)} - \frac{1}{(x + 2) (x - 2)} + \frac{x^{2} - 2 x - 35 + 5}{(x + 2) (x - 2) (x + 3) (x - 4)} \\ = \frac{1}{(x + 2) (x + 3)} - \frac{1}{(x + 2) (x - 2)} + \frac{x^{2} - 2 x - 30}{(x + 2) (x - 2) (x + 3) (x - 4)} \end{aligned}

::=1(x+2)(x+2)(x+3)-1(x+2)(x-2)+2)+x2-2-2(x+2)(x-2)(x+3)(x-4)=1(x+2)(x+3)(x+3)=1(x+2(x+2)(x-2)(x-2)+x2-2(x-2)(x-2)(2)+x2-x30(x+2)(x-2)(x+3)(x-4))

Note that the right expression has 4 factors in the denominator while each of the left expressions have two that match and two that are missing from those four factors. This tells you what you need to multiply each expression by in order to have the denominators match up.
::请注意, 右表达式在分母中有 4 个因素, 而左表达式中的每个表达式都有 2 个匹配, 4 个因素中缺少 2 个缺失。这表示您需要将每个表达式乘以什么才能使分母匹配。

$\begin{aligned} = \frac{(x - 2) (x - 4)}{(x + 2) (x - 2) (x + 3) (x - 4)} - \frac{(x + 3) (x - 4)}{(x + 2) (x - 2) (x + 3) (x - 4)} + \frac{x^{2} - 2 x - 30}{(x + 2) (x - 2) (x + 3) (x - 4)} \end{aligned}$
::= = (x- 2)(x- 2)(x+2)(x-2)(x+3)(x-3)(x-4)(x+3)(x-4)(x+3)(x-2)(x-2)(x-2)(x-2)(x-4)+x2-2(x-4)(x-2)(x-2)(x-2)(x-2)(x-2)(x+3)(x-4)

Now since the rational expressions have a common denominator, the numerators may be multiplied out and combined. Sometimes instead of rewriting an expression repeatedly in mathematics you can use an abbreviation. In this case, you can replace the denominator with the letter $\begin{aligned} D \end{aligned}$ and then replace it at the end.
::既然理性表达式有一个共同的分母, 分子可能会被乘出并合并。有时, 您可以使用缩写, 而不是在数学中反复重写一个表达式。在这种情况下, 您可以用字母 D 替换该分母, 然后在结尾处替换它。

\begin{aligned} = \frac{(x - 2) (x - 4) - (x + 3) (x - 4) + x^{2} - 2 x - 30}{D} \\ = \frac{x^{2} - 6 x + 8 - [x^{2} - x - 12] + x^{2} - 2 x - 30}{D} \end{aligned}

::=(x-2)(x-4)-(x+3)(x-4)+x2-2-2x-30D=x2-6x+

x2-x-12]+x2-2-2x-30D)

Notice how it is extremely important to use brackets to indicate that the subtraction applies to all the terms of the middle expression not just $\begin{aligned} x^{2} \end{aligned}$ . This is one of the most common mistakes.
::注意使用括号表示减法适用于中间表达式中的所有术语,而不仅仅是 x2, 这一点极为重要。这是最常见的错误之一。

\begin{aligned} = \frac{x^{2} - 6 x + 8 - x^{2} + x + 12 + x^{2} - 2 x - 30}{D} \\ = \frac{x^{2} - 7 x - 10}{D} \end{aligned}

::=x2-6x+8-x2+x12+12+x2-2x-30D=x2-7x-10D

Now replace $\begin{aligned} D . \end{aligned}$
::现在取代D。

\begin{aligned} \frac{x^{2} - 7 x - 10}{(x - 2) (x + 3) (x - 4)} \end{aligned}

::x2-7x-10(x-2)(x+3)(x-4)

Example 5
::例5

Simplify the following expression.
::简化以下表达式。

\begin{aligned} \frac{\frac{1}{x + 1} - \frac{1}{x + 2}}{\frac{1}{x - 2} - \frac{1}{x + 1}} \end{aligned}

The expression itself does not look like a rational expression, but it can be rewritten so it is more recognizable. Also, working with fractions within fractions is an important skill.
::1x+1-1-1x+21x-2-2-1x+1 表达式本身看起来不象一个理性表达式,但可以重写,使其更容易识别。此外,使用分数中的分数工作是一项重要的技能。

\begin{aligned} = (\frac{1}{x + 1} - \frac{1}{x + 2}) \div (\frac{1}{x - 2} - \frac{1}{x + 1}) \\ = [\frac{(x + 2)}{(x + 1) (x + 2)} - \frac{(x + 1)}{(x + 1) (x + 2)}] \div [\frac{(x + 1)}{(x + 1) (x - 2)} - \frac{(x - 2)}{(x + 1) (x - 2)}] \\ = [\frac{1}{(x + 1) (x + 2)}] \div [\frac{3}{(x + 1) (x - 2)}] \\ = \frac{1}{(x + 1) (x + 2)} \cdot \frac{(x + 1) (x - 2)}{3} \\ = \frac{(x - 2)}{3 (x + 2)} \end{aligned}

::= = 1x+1 - 1x+2) (1x-2 - 2 - 1x+1) = [(x+2)(x+1)(x+1)(x+2)(x+1)(x+1)(x+1)(x+1)(x-2) [(x+1)(x+1)(x-2) = [1(x+1)(x+1)(x+1)(x-2) = [3(x+1)(x-2) = 1(x+1)(x+2)(x+1)(x+1)(x+2) = (x+1)(x+1)(x-2) 3= (x-2) 3(x+2) = (x+2)

Bonus Example
::奖金实例

Simplify the following expression which has an infinite number of fractions nested within other fractions.
::简化以下表达式, 该表达式有无限数量的分数嵌入其他分数中。

\begin{aligned} \frac{- 1}{2 + \frac{- 1}{2 + \frac{- 1}{2 + \frac{- 1}{2 + \frac{- 1}{2 + \frac{- 1}{2 + \dots}}}}}} \end{aligned}

It would be an exercise in futility to attempt to try to compute this expression directly. Instead, notice that the repeating nature of the expression lends itself to an extremely nice substitution.
::试图直接计算这一表达方式是徒劳的。相反,应注意该表达方式的重复性质本身就是一种极好的替代。

Let $\begin{aligned} \frac{- 1}{2 + \frac{- 1}{2 + \frac{- 1}{2 + \frac{- 1}{2 + \frac{- 1}{2 + \frac{- 1}{2 + \dots}}}}}} = x \end{aligned}$
::让- 12 @ 12 @ 12 @ 12+... =x

Notice that the red portion of the fraction is exactly the same as the rest of the fraction and so $\begin{aligned} x \end{aligned}$ may be substituted there and solved.
::请注意,分数的红色部分与分数的其余部分完全相同,因此,x可以在那里替换并解决。

\begin{aligned} \frac{- 1}{2 + x} & = x \\ - 1 & = x (2 + x) \\ - 1 & = x^{2} + 2 x \\ 0 & = x^{2} + 2 x + 1 \\ 0 & = (x + 1)^{2} \\ x & = - 1 \end{aligned}

::- 12+x=x - 1=x(2+x) - 1=x2+2x0=x2+2xx+10=(x+1)2x*1

The reason why this expression is included in this concept is because it highlights one problem solving aspect of simplifying expressions that distinguishes PreCalculus from Algebra 1 and Algebra 2.
::之所以将这一表达方式列入这一概念,是因为它突出了简化表达方式的一个解决问题的方面,即区分代数1和代数2和前代数1和代数2。

Summary

A rational expression is a ratio between two expressions.
::理性表达式是两个表达式之间的比例。

To simplify rational expressions, factor each polynomial and cancel identical factors in the numerator and denominator.
::为了简化理性表达方式,请将每个多数值乘以系数,并取消分子和分母中的相同系数。

To add or subtract rational expressions, find the least common denominator, and multiply each expression by the common denominator over itself before combining numerators.
::要添加或减去理性表达式,要找到最小的公分母,并在合并数字器之前将每个表达式乘以共同的分母。

To multiply rational expressions, write the product of all numerator factors over the product of all denominator factors, then cancel identical factors.
::要乘以理性表达式,请在所有分母因素的产物上写出所有分子系数的产物,然后取消相同的因数。

To divide rational expressions, rewrite the division problem as a multiplication problem by multiplying the first rational expression by the reciprocal of the second, then follow the steps for multiplying rational expressions.
::要将合理表达方式分开, 将分裂问题重写为乘法问题, 将第一个合理表达方式乘以第二个合理表达方式乘以第二个合理表达方式乘以第二个合理表达方式, 然后按步骤将合理表达方式乘以多个步骤。

Review
::回顾

Perform the indicated operation and simplify as much as possible.
::执行指定的操作并尽可能简化。

$\begin{aligned} \frac{x^{2} + 5 x + 4}{x^{2} + 4 x + 3} \cdot \frac{5 x^{2} + 15 x}{x + 4} \end{aligned}$
::x2+5x+4x2+4x+5x2+5x2+15xxx+4

$\begin{aligned} \frac{x^{2} - 4}{x^{2} + 4 x + 4} \cdot \frac{7}{x - 2} \end{aligned}$
::x2 - 4x2+4x+4=7x-2

$\begin{aligned} \frac{4 x^{2} - 12 x}{5 x + 10} \cdot \frac{x + 2}{x} \div \frac{x - 3}{1} \end{aligned}$
::4x2 - 12x5x+10xx+2x_ x- 31

$\begin{aligned} \frac{4 x^{3} - 4 x}{x} \div \frac{2 x - 2}{x - 4} \end{aligned}$
::4x3-4xxx2x-2x-4

$\begin{aligned} \frac{2 x^{3} + 8 x}{x + 1} \div \frac{x}{2 x^{2} - 2} \end{aligned}$
::2x3+8xxx+1x2x2-2

$\begin{aligned} \frac{3 x - 1}{x^{2} + 2 x - 15} - \frac{2}{x + 5} \end{aligned}$
::3x-1x2+2x-15-2x+5

$\begin{aligned} \frac{x^{2} - 8 x + 7}{x^{2} - 4 x - 21} \cdot \frac{x^{2} - 9}{1 - x^{2}} \end{aligned}$
::x2-8x+7x2-4x-21x2-91-x2

$\begin{aligned} \frac{2}{x + 7} + \frac{1}{x - 7} \end{aligned}$
::2x+7+1x-7

$\begin{aligned} \frac{6}{x - 7} - \frac{6}{x + 7} \end{aligned}$
::6 - 7 - 6x+7 6 - 7 - 6x+7

$\begin{aligned} \frac{3 x + 35}{x^{2} - 25} + \frac{2}{x + 5} \end{aligned}$
::3x+35x2-25+2x+5

$\begin{aligned} \frac{2 x + 20}{x^{2} - 4 x - 12} + \frac{2}{x + 2} \end{aligned}$
::2x+20x2-4x-12+2x+2

$\begin{aligned} \frac{2}{x + 6} - \frac{x - 9}{x^{2} - 3 x - 18} \end{aligned}$
::2x+6-x-9x2-3x-18

$\begin{aligned} - \frac{5 x + 30}{x^{2} + 11 x + 30} + \frac{2}{x + 5} \end{aligned}$
::-5x+30x2+11x+30+2x+5

$\begin{aligned} \frac{x + 3}{x + 2} + \frac{x^{3} + 4 x^{2} + 5 x + 20}{2 x^{4} + 2 x^{2} - 40} \end{aligned}$
::x+3x+2+3x3+4x2+5x+202x4+2x2-40
$\begin{aligned} \frac{- 4}{2 + \frac{- 4}{2 + \frac{- 4}{2 + \frac{- 4}{2 + \frac{- 4}{2 + \frac{- 4}{2 + \dots}}}}}} \end{aligned}$

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

Working with Rational Expressions ::与有理表达式一起工作

Adding and Subtracting Rational Expressions ::添加和减减理性表达式

Multiplying and Dividing Rational Expressions ::乘法和分裂逻辑表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

= 2 ( x + 2 ) ( x − 2 ) 2 ( x + 2 ) ( x + 3 ) − ( x 2 + 8 ) 2 ( x + 2 ) ( x + 3 ) = 2 ( x 2 − 4 ) − x 2 − 8 2 ( x + 2 ) ( x + 3 ) = x 2 − 16 2 ( x + 2 ) ( x + 3 ) Example 3 ::=2x+2(x+2)(x-2)(x+2)(x+3)-(x2+8)(x2)(x+3)=2x2-4)(x2-82(x+2)(x+3)(x+3)=x2-162(x+2)(x+3)例3

Example 4 ::例4

Example 5 ::例5

Bonus Example ::奖金实例

Review ::回顾

Review (Answers) ::回顾(答复)