Section: 简化逻辑表达式 | 9.2 简化理性表达式

Section outline

A farmer can model the costs of producing x bushels of corn by $\begin{align*}3.5x^2+500x\end{align*}$ . To find the average cost of producing bushels of corn, we can divide the total cost by the number of bushels. What expression models the average cost of a bushel of corn? To determine this, we will need to know how to simplify rational expressions, and we discuss that in this section.
::农民可以以3.5x2+500x来模拟生产玉米的x灌木的成本。为了找到生产玉米灌木的平均成本,我们可以将总成本除以灌木的数量。哪种表达方式可以以玉米灌木的平均成本为模型? 要确定这一点,我们需要知道如何简化合理表达方式,我们在本节讨论这一点。

Rational Expressions
::理性表达式

In this chapter, we will consider rational functions . A rational function is a function , $\begin{align*}f(x)\end{align*}$ , such that $\begin{align*}f(x)=\frac{p(x)}{q(x)}\end{align*}$ , where $\begin{align*}p(x)\end{align*}$ and $\begin{align*}q(x)\end{align*}$ are both . Before we consider the functions, we will first work with rational expressions , expressions of the form $\begin{align*}\frac{p(x)}{q(x)}\end{align*}$ . Rational expressions are examples of fractional expressions , which are expressions in which any algebraic expression is divided by any algebraic expression.
::在本章中, 我们将考虑合理的函数。理性函数是一个函数 f( x), 例如 f( x) = p( x) q( x) , 其中 p( x) 和 q( x) 两者都是 p( x) 和 q( x) 。在我们考虑函数之前, 我们将首先使用理性表达式, 表达式 p( x) q( x) 。理性表达式是分数表达式的示例, 即任何代数表达式被任何代数表达式除以的表达式。

Example 1
::例1

Determine whether the following are rational expressions: $\begin{align*}\text{a.} \ \frac{\text{-}4x^3+6x}{5x^2+1}, \ \text{b.}\ \frac{\sqrt{2x+3}}{x}, \ \text{c.} \ \frac{4}{\text{-}0.75x^2+2}, \ \text{d.} \ \frac{8x^{\text{-}1}}{|x|} \end{align*}$ .
::确定以下是否为合理表达式: a. - 4x3+6x5x2+1, b. 2x+3x, c. 4- 0. 75x2+2, d. 8x-1x* 。

Solution:
::解决方案 :

a. $\begin{align*}\frac{\text{-}4x^3+6x}{5x^2+1}\end{align*}$ is a rational expression . Both the numerator and the denominator are polynomials.
::a. -4x3+6x5x2+1 是一个合理的表达式。分子和分母都是多分子。

b. $\begin{align*}\frac{\sqrt{2x+3}}{x}\end{align*}$ is not a rational expression. The numerator involves a square root , which cannot be part of a polynomial .
::b. 2x+3x 不是一个合理的表达式。分子包含一个正方根,不能作为多面体的一部分。

c. $\begin{align*}\frac{4}{\text{-}0.75x^2+2}\end{align*}$ is a rational expression. The numerator is a degree-zero polynomial. The denominator has real coefficients and non-negative integer exponents, so it is also a polynomial.
::c. 4- 0. 75x2+2是一个合理的表达式。分子是一个度为零的多元分子。分母有真实的系数和非负的整数指数,因此它也是一个多元分子。

d. $\begin{align*}\frac{8x^{\text{-}1}}{|x|} \end{align*}$ is not a rational expression. The numerator includes a negative exponent and the denominator includes absolute value . Neither can be part of a polynomial.
::d. 8x-1x不是理性表达式。分子包括负引号,分母包括绝对值,也不能作为多元值的一部分。

Simplifying Rational Expressions
::简化逻辑表达式

Like any fraction , a rational expression can be simplified. Simplifying a rational expression, we use the same process we follow to simplify fractions with numbers: To factor the polynomials, determine if any factors are the same, and then cancel out any common factors. Compare these processes below .
::与任何分数一样, 一个理性表达式可以简化。简化一个理性表达式, 我们使用我们所遵循的相同程序来简化带有数字的分数 : 要将多数值因素考虑在内, 确定任何因素是否相同, 然后取消任何共同因素。比较下面的这些进程。

Fraction: $\begin{align*}\frac{9}{15} = \frac{\cancel{3} \cdot 3}{\cancel{3} \cdot 5} = \frac{3}{5}\end{align*}$
::分数: 915=333=35

Rational Expression: $\begin{align*}\frac{x^2+6x+9}{x^2+8x+15} = \frac{\cancel{(x+3)}(x+3)}{\cancel{(x+3)}(x+5)} = \frac{x+3}{x+5}\end{align*}$
::有理表达式: x2+6x+9x2+8x+8x+15=(x+3)(x+3)(x+3)(x+3)(x+5)=x+3x+5

With both fractions, we factored the numerator and denominator into prime factors. Then we canceled the common factors.
::使用两个分数, 我们将分子和分母乘以质因子。然后我们取消了共同因数。

To simplify rational expressions, factor the numerator and denominator and cancel the common factors. In general, for any expressions A, B, and C:
::为简化理性表达方式,乘以分子和分母,并取消共同因素。一般而言,对于任何表达式A、B和C,

$\begin{aligned} \frac{A C}{B C} = \frac{A}{B} . \end{aligned}$
$\begin{align*}\frac{AC}{BC}=\frac{A}{B}.\end{align*}$
::ACBC=AB(AB) (AB) (ABBC=AB) (AB) (ABBC=AB) (AB) (ABBC=AB) (AB) (ABBC=AB) (AB) (ABBC=AB) (AB) (ABBC=AB) (AB) (ABBC=(AB) (AB) (AB) (ABBC=(AB) (AB) (ABB) (ABB) (ABBC=(AB) (AB) (AB) (ABBC=(AB) (AB) (ABB) (ABC) (ABC) (ABC=(AB) (AB) (ABC) (ABC=(AB) (AB) (AB) (AB)。

Example 2
::例2

A farmer can model the costs of producing x bushels of corn by $\begin{align*}3.5x^2+500x\end{align*}$ . To find the average cost of producing x bushels of corn, we can divide the total cost by the number of bushels. What expression models the average cost of a bushel of corn?
::农民可以以3.5x2+500x来模拟生产玉米的x灌木的成本。要找到生产玉米的x灌木的平均成本,我们可以将总成本除以灌木的数量。什么表达方式可以以玉米灌木的平均成本为模型?

Solution: The expression that models the average cost is $\begin{align*}\frac{3.5x^2+500x}{x}\end{align*}$ , the total cost divided by the number of bushels. To simplify this, we start by factoring the numerator:
::解决方案: 平均成本为3.5x2+500xx的模型表达式, 总成本除以灌木数量。为简化此选项, 我们首先对分子进行计数 :

$\begin{aligned} \frac{3.5 x^{2} + 500 x}{x} = \frac{x (3.5 x + 500)}{x} = 3.5 x + 500. \end{aligned}$
$\begin{align*}\frac{3.5x^2+500x}{x}=\frac{\cancel{x}(3.5x+500)}{\cancel{x}}=3.5x+500.\end{align*}$
::3.5x2+500xx=x(3.5x+500)x=3.5x+500。

On average, it costs $\begin{align*}3.5x+500\end{align*}$ dollars to produce a bushel of corn.
::平均而言,生产一棵玉米树的成本为3.5x+500美元。

by The Farmer's Life demonstrates how a farmer values one ear of corn.
::农民生命展示了农民如何重视一耳玉米。

Example 3
::例3

Simplify $\begin{align*}\frac{3x^2-x}{3x^2}\end{align*}$ .
::简化 3x2 - x3x2 。

Solution: Factor the numerator and denominator, then cancel any common factors with the denominator.
::解答:乘以分子和分母,然后用分母取消任何共同因素。

$\begin{aligned} \frac{3 x^{2} - x}{3 x^{2}} = \frac{x (3 x - 1)}{3 \cdot x \cdot x} = \frac{3 x - 1}{3 x} \end{aligned}$
$\begin{align*}\frac{3x^2-x}{3x^2} = \frac{\cancel{x}(3x-1)}{3 \cdot \cancel{x} \cdot x} = \frac{3x-1}{3x}\end{align*}$
::3x2-x3x2=x(3x-1)3xxxx=3x-13x

Notice we do not cancel the $\begin{align*}3x\end{align*}$ terms . The operation in the numerator is subtraction , not multiplication , so we cannot cancel.
::注意我们不取消 3x 条件。分子中的操作是减法, 不是乘法, 所以我们不能取消。

by Mathispower4u demonstrates how to simplify rational expressions.
::由 Mathispower4u 演示如何简化理性表达方式。

Example 4
::例4

Simplify $\begin{align*}\frac{2x^3}{4x^2-6x}\end{align*}$ .
::简化 2x34x2 - 6x 。

Solution: The numerator factors as $\begin{align*}2x^3=2 \cdot x \cdot x \cdot x,\end{align*}$ and the denominator is $\begin{align*}4x^2-6x=2x(2x-3)\end{align*}$ .
::解析度:2x3=2xxxxxxxxx,分母为4x2-6x=2x(2x-3)。

$\begin{aligned} \frac{2 x^{3}}{4 x^{2} - 6 x} = \frac{2 \cdot x \cdot x \cdot x}{2 \cdot x \cdot (2 x - 3)} = \frac{x^{2}}{2 x - 3} \end{aligned}$
$\begin{align*}\frac{2x^3}{4x^2-6x} = \frac{\cancel{2} \cdot \cancel{x} \cdot x \cdot x}{\cancel{2} \cdot \cancel{x} \cdot(2x-3)} = \frac{x^2}{2x-3}\end{align*}$
::2x34x2-6x=2xxxxxxxxxx2xxxx}(2x-3)=x22x-3

Example 5
::例5

Simplify $\begin{align*}\frac{x^2-6x-27}{2x^2-19x+9}\end{align*}$ .
::简化 x2 - 6x - 272x2 - 19x+9 。

Solution: Factor both the numerator and the denominator to see if there are any common factors.
::解决办法:将分子和分母都考虑在内,以确定是否存在任何共同因素。

$\begin{aligned} \frac{x^{2} - 6 x - 27}{2 x^{2} - 19 x + 9} = \frac{(x - 9) (x + 3)}{(x - 9) (2 x - 1)} = \frac{x + 3}{2 x - 1} \end{aligned}$
$\begin{align*}\frac{x^2-6x-27}{2x^2-19x+9} = \frac{\cancel{(x-9)}(x+3)}{\cancel{(x-9)}(2x-1)} = \frac{x+3}{2x-1}\end{align*}$
::x2 - 6x - 272x2-19x+9=(x- 9)(x+3)(x- 9)(2x-1)=x+32x-1

Example 6
::例6

Simplify $\begin{align*}\frac{6x^2-7x-3}{2x^3-3x^2}\end{align*}$ .
::简化 6x2 - 7x - 32x3 - 3x2 。

Solution: F actor the numerator and find the GCF of the denominator. Cancel the common factors.
::解决方案: 乘以分子并找到分母的全球合作框架。取消共同因素。

$\begin{aligned} \frac{6 x^{2} - 7 x - 3}{2 x^{3} - 3 x^{2}} = \frac{(2 x - 3) (3 x + 1)}{x^{2} (2 x - 3)} = \frac{3 x + 1}{x^{2}} \end{aligned}$
$\begin{align*}\frac{6x^2-7x-3}{2x^3-3x^2} = \frac{\cancel{(2x-3)}(3x+1)}{x^2\cancel{(2x-3)}} = \frac{3x+1}{x^2}\end{align*}$
::6x2-7x-32x3-3x2=(2x-3)(3x+1)xxx2(2x-3)=3x+1x2x2

Example 7
::例7

Simplify $\begin{align*}\frac{x^3-4x}{x^5+4x^3-32x}\end{align*}$ .
::简化 x3 - 4x25+4x3 - 32x

Solution: First we need to factor out the GCF, and then we can factor the remaining quadratic polynomials .
::解决办法:首先,我们需要将绿色气候基金考虑在内,然后,我们可以将剩余的四边多面体因素考虑在内。

$\begin{aligned} \frac{x^{3} - 4 x}{x^{5} + 4 x^{3} - 32 x} = \frac{x (x^{2} - 4)}{x (x^{4} + 4 x^{2} - 32)} = \frac{x (x - 2) (x + 2)}{x (x^{2} - 4) (x^{2} + 8)} = \frac{x (x - 2) (x + 2)}{x (x - 2) (x + 2) (x^{2} + 8)} = \frac{1}{x^{2} + 8} \end{aligned}$
$\begin{align*}\frac{x^3-4x}{x^5+4x^3-32x} = \frac{x(x^2-4)}{x(x^4+4x^2-32)} = \frac{x(x-2)(x+2)}{x(x^2-4)(x^2+8)} = \frac{\cancel{x (x-2)(x+2)}}{\cancel{x (x-2)(x+2)}(x^2+8)} = \frac{1}{x^2+8}\end{align*}$
::x3-4x5+4x5+4x3-32x=xx(x2-4)x(x4+4x2)x(x4+4x2-2-32)x(x-2)(x+2-2)x(x2-4(x2)+8)=x(x-2)(x+2)(x+2)xx(x-2)(x+2)(x+2)(x2+8)=1x2+8

Example 8
::例8

Simplify $\begin{align*}\frac{3-x}{6x-18}\end{align*}$ .
::简化 3- x6x-18 。

Solution: The numerator is already prime. Let's factor the denominator.
::解答: 分子已经是质数。让我们来乘以分母。

$\begin{aligned} \frac{3 - x}{6 x - 18} = \frac{3 - x}{6 (x - 3)} \end{aligned}$
$\begin{align*}\frac{3-x}{6x-18}=\frac{3-x}{6(x-3)}\end{align*}$
::3-x6x-18=3-x6(x-3)

Notice the binomial in the numerator and the binomial in the denominator have the same terms, but the signs are off. If we factor $\begin{align*}\text{-}1\end{align*}$ out of the numerator, we will have a common factor to cancel.
::注意分子中的二进制和分母中的二进制都有相同的条件, 但符号关闭了。如果我们从分子中乘以 - 1, 我们将会有一个共同的取消因素。

$\begin{aligned} \frac{3 - x}{6 (x - 3)} = \frac{- 1 (- 3 + x)}{6 (x - 3)} = \frac{- 1 (x - 3)}{6 (x - 3)} = - \frac{1}{6} \end{aligned}$
$\begin{align*}\frac{3-x}{6(x-3)}=\frac{\text{-}1(\text{-}3+x)}{6(x-3)}=\frac{\text{-}1 \cancel{(x-3)}}{6 \cancel{(x-3)}}=\text{-}\frac{1}{6}\end{align*}$
::3-x6(x-3)=-1(-3+x)6(x-3)=-1(x-3)6(x-3)=-1(x-3)6(x-3)=-16

Example 9
::例9

Simplify $\begin{align*}\frac{x^2+6x+8}{x^2+6x+9}\end{align*}$ .
::简化 x2+6x+8x2+6x+9。

Solution:
::解决方案 :

$\begin{aligned} \frac{x^{2} + 6 x + 8}{x^{2} + 6 x + 9} = \frac{(x + 4) (x + 2)}{(x + 3) (x + 3)} \end{aligned}$
$\begin{align*}\frac{x^2+6x+8}{x^2+6x+9} = \frac{(x+4)(x+2)}{(x+3)(x+3)}\end{align*}$
::x2+6x+8x2+6x6x+9=(x+4)(x+2)(x+3)(x+3)

There are no common factors, so this is the simplest form .
::没有共同的因素,所以这是最简单的形式。

by Randy Anderson demonstrates how to simplify rational expressions.
::Randy Anderson展示了如何简化理性表达方式。

WARNING
::警告

$\begin{align*}\frac{x+3}{x+5}\end{align*}$    is completely factored. Do not cancel out the $\begin{align*}x\end{align*}$ ’s.
$\begin{align*}\frac{3x}{5x}\end{align*}$ simplifies to $\begin{align*}\frac{3}{5}\end{align*}$ , but $\begin{align*}\frac{x+3}{x+5}\end{align*}$    does not because of the addition sign.
To prove this, we will plug in a number for $\begin{align*}x\end{align*}$    to and show that the fraction does not simplify to    $\begin{align*}\frac{3}{5}\end{align*}$ . If $\begin{align*}x=2\end{align*}$ , then $\begin{align*}\frac{2+3}{2+5} = \frac{5}{7} \ne \frac{3}{5}\end{align*}$ .
::x+3x+5 已被完全计算。不要取消 x。 3x5x 简化为35, 但由于添加符号, x+3x+5 并不代表此特性。为了证明这一点, 我们将插入 x 的编号, 并显示该分数不简化为35。如果 x=2, 那么 2+32+5=5735 , 则该分数不简化为35 。

You can only cancel one factor for one factor, not all factors that are the same. The following is incorrect: $\begin{align*}\frac{(x+1)(x+2)}{(x+1)^2} \ne \frac{\cancel{(x+1)}(x+2)}{\cancel{(x+1)^2}} = x+2.\end{align*}$
::您只能取消一个因数的一个因数, 而不是所有相同的因数。以下不正确 : (x+1)(x+2)(x+1)(x+1)(x+1)(x+2)(x+2)(x+2)(x+1)(x+1)(x+1)(x+1)(x+1)2=x+2) 。

You can cancel one factor from the numerator and one from the denominator, not two from the numerator or denominator. The following is incorrect: $\begin{align*}\frac{(x+2)(x+3)(x+2)}{x+4} \ne \frac{\cancel{(x+2)}(x+3)\cancel{(x+2)}}{x+4}=\frac{x+3}{x+4}.\end{align*}$
::您可以从分子中取消一个因数, 从分母中取消一个因数, 而不是从分子或分母中取消两个因数。以下不正确 : (x+2)(x+3)(x+2)(x+2)(x+2)x+4}(x+2(x+3)(x+2)x+4=x+3x+4) 。

Summary
::摘要

To simplify rational expressions, factor the polynomials in the numerator and denominator, and then cancel any common factors.
::为了简化理性表达方式,在分子和分母中考虑到多数值,然后取消任何共同因素。

Review
::回顾

Simplify the following rational expressions:
::简化以下合理表达式:

1. $\begin{align*}\frac{4x^3}{2x^2+3x}\end{align*}$
::1. 4x32x2+3x

2. $\begin{align*}\frac{x^3+x^2-2x}{x^4+4x^3-5x^2}\end{align*}$
::2. x3+x2-2x4+4x3-5x2

3. $\begin{align*}\frac{2x^2-5x-3}{2x^2-7x-4}\end{align*}$
::3. 2x2 - 5x - 32x2 - 7x - 4

4. $\begin{align*}\frac{5x^2+37x+14}{5x^3-33x^2-14x}\end{align*}$
::4. 5x2+37x+145x3-33x2-14x

5. $\begin{align*}\frac{8x^2-60x-32}{\text{-}4x^2+26x+48}\end{align*}$
::5. 8x2-60x-32-4x2+26x+48

6. $\begin{align*}\frac{6x^3-24x^2+30x-120}{9x^4+36x^2-45}\end{align*}$
::6. 6x3-24x2+30x-12009x4+36x2-45

7. $\begin{align*}\frac{6x^2+5x-4}{6x^2-x-1}\end{align*}$
::7. 6x2+5x-46x2-x-1

8. $\begin{align*}\frac{x^4+8x}{x^4-2x^3+4x^2}\end{align*}$
::8. x4+8x24-2x3+4x2

9. $\begin{align*}\frac{6x^4-3x^3-63x^2}{12x^2-84x}\end{align*}$
::9. 6x4-3x3-63x212x2-84x

10. $\begin{align*}\frac{x^5-3x^3-4x}{x^4+2x^3+x^2+2x}\end{align*}$
::10. x5-3x3-4x4+2x3+x2+2x2+2xx

11. $\begin{align*}\frac{\text{-}3x^2+25x-8}{x^3-8x^2+x-8}\end{align*}$
::11. -3x2+25x-8x3-8x2+x-8

12. $\begin{align*}\frac{\text{-}x^3+3x^2+13x-15}{\text{-}2x^3+7x^2+20x-25}\end{align*}$
::12. -x3+3x2+13x-15-2x3+7x2+20x-25

Explore More
::探索更多

1. Does $\begin{align*}\frac{x-2}{x-6}\end{align*}$ simplify to $\begin{align*}\frac{1}{3}\end{align*}$ ? Explain why or why not. Does $\begin{align*}\frac{5x}{10x}\end{align*}$ simplify to $\begin{align*}\frac{1}{2}\end{align*}$ ? Explain why or why not. In your own words, explain the difference between the previous two expressions and why one simplifies and one does not.
::1. x-2x-6 是否简化为 13 ? 解释原因或原因; 5x10x 简化为 12 ? 解释原因或原因; 用您自己的语言解释前两个表达式的区别,为什么一个简化而一个不简化。

2. The width of a rectangle is $\begin{align*}6x + 8\end{align*}$ , and the length of the rectangle is $\begin{align*}12x + 16\end{align*}$ . Determine the ratio of the width to the perimeter.
::2. 矩形宽度为6x+8,矩形长为12x+16。确定宽度与周边之比。

3. An isosceles triangle has two sides of length $\begin{align*}9x+3\end{align*}$ . The perimeter of the triangle is $\begin{align*}30x+10\end{align*}$ . Determine the ratio of the perimeter to the third side .
::3. 等分形三角形的两面长度为9x+3。三角形的周边为30x+10。确定周边与第三侧的比例。

4. The area of a rectangle is $\begin{align*}2x^4 - 2\end{align*}$ . The width of the rectangle is $\begin{align*}x^2 + 1\end{align*}$ . What is the length of the rectangle?
::4. 矩形区域为 2x4-2。矩形宽度为 x2+1。矩形的长度是多少?

5. Error Analysis: Identify the error in Student A's work. Explain what the problem is.
::5. 错误分析:查明学生A工作中的错误,解释问题是什么。

$\begin{aligned} \frac{x^{2} - 2 x + 3}{x^{2} + 8 x + 12} = \frac{x^{2} - 2 x + 3}{x^{2} + 8 x + 12} = \frac{- 2 x + 3}{8 x + 12} \end{aligned}$
$\begin{align*}\frac{x^2-2x+3}{x^2+8x+12}=\frac{\cancel{x^2}-2x+3}{\cancel{x^2}+8x+12}=\frac{\text{-}2x+3}{8x+12}\end{align*}$
::x2 - 2x+3x2+8x+12=x2 -2x+3x2+8x+12=2x+38x+12

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section.
::尝试一下这种互动关系,加强本节所探讨的概念。

Section outline

Rational Expressions ::理性表达式

Example 1 ::例1

Simplifying Rational Expressions ::简化逻辑表达式

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Example 7 ::例7

Example 8 ::例8

Example 9 ::例9

WARNING ::警告

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

Rational Expressions
::理性表达式

Example 1
::例1

Simplifying Rational Expressions
::简化逻辑表达式

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Example 7
::例7

Example 8
::例8

Example 9
::例9

WARNING
::警告

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX