Course: 9.7 简化理性表达式

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

Rational Expressions
::理性表达式

Recall that a rational function is a function , $\begin{aligned} f (x) \end{aligned}$ , such that $\begin{aligned} f (x) = \frac{p (x)}{q (x)} \end{aligned}$ , where $\begin{aligned} p (x) \end{aligned}$ and $\begin{aligned} q (x) \end{aligned}$ are both . A rational expression , is just $\begin{aligned} \frac{p (x)}{q (x)} \end{aligned}$ . Like any fraction , a rational expression can be simplified. To simplify a rational expression, you will need to factor the polynomials, determine if any factors are the same, and then cancel out any like factors.
::回顾理性函数是一个函数 f(x) , 即 f(x) = p(x) = p(x) 和 q(x) , p(x) 和 q(x) 是两者的函数。理性表达只是 p(x) q(x) 。与任何部分一样, 理性表达也可以简化。要简化理性表达, 您需要将多元表达法计算在内, 确定是否有任何因素相同, 然后取消任何类似因素。

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

Rational Expression: $\begin{aligned} \frac{x^{2} + 6 x + 9}{x^{2} + 8 x + 15} = \frac{(x + 3) (x + 3)}{(x + 3) (x + 5)} = \frac{x + 3}{x + 5} \end{aligned}$
::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 broke apart the numerator and denominator into the prime factorization. Then, we canceled the common factors.
::以两个分数, 我们打破了分子和分母进入质因子化。然后, 我们取消了共同因素。

Important Note: $\begin{aligned} \frac{x + 3}{x + 5} \end{aligned}$ is completely factored. Do not cancel out the $\begin{aligned} x \end{aligned}$ ’s! $\begin{aligned} \frac{3 x}{5 x} \end{aligned}$ reduces to $\begin{aligned} \frac{3}{5} \end{aligned}$ , but $\begin{aligned} \frac{x + 3}{x + 5} \end{aligned}$ does not because of the addition sign. To prove this, we will plug in a number for $\begin{aligned} x \end{aligned}$ to and show that the fraction does not reduce to $\begin{aligned} \frac{3}{5} \end{aligned}$ . If $\begin{aligned} x = 2 \end{aligned}$ , then $\begin{aligned} \frac{2 + 3}{2 + 5} = \frac{5}{7} \neq \frac{3}{5} \end{aligned}$ .
::重要注意 : x+3x+5 已完全计算在内。不要取消 x 的 3x5 。 3x5x 减为35, 但 x+3x+5 并不是因为添加符号。为了证明这一点, 我们将插入 x 的编号, 并显示如果 x=2, 那么 2+32+5=57QQQ35, 则该分数不会减为35 。

Let's simplify the following expressions.
::让我们简化以下表达式。

$\begin{aligned} \frac{2 x^{3}}{4 x^{2} - 6 x} \end{aligned}$
::2x34x2-6x

The numerator factors to be $\begin{aligned} 2 x^{3} = 2 \cdot x \cdot x \cdot x \end{aligned}$ and the denominator is $\begin{aligned} 4 x^{2} - 6 x = 2 x (2 x - 3) \end{aligned}$ .
::2x3=2xxxxxxxxx和分母为4x2-6x=2x(2x-3)。

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

$\begin{aligned} \frac{6 x^{2} - 7 x - 3}{2 x^{3} - 3 x^{2}} \end{aligned}$
::6x2 - 7x - 32x3 - 3x2

Factor the numerator and find the GCF of the denominator and cancel out the like terms .
::乘以分子数,找到分母的绿色气候基金,取消类似条件。

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

$\begin{aligned} \frac{x^{2} - 6 x + 27}{2 x^{2} - 19 x + 9} \end{aligned}$
::x2-6x+272x2-19x+9

Factor both the top and bottom and see if there are any common factors.
::将上部和下部都考虑在内,看看是否存在任何共同因素。

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

Special Note: Not every polynomial in a rational function will be factorable. Sometimes there are no common factors. When this happens, write “not factorable.”
::特殊注释 : 理性函数中并不是每个多边函数都是可以考虑的因素。有时没有共同的因素。当发生这种情况时, 请写“ 不可考虑的因素 ” 。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the length of the rectangle.
::早些时候,有人要求你找到矩形的长度。

Recall that the the area of a rectangle is the length times the width. To find the length, we can therefore divide the area by the width. So we're looking for $\begin{aligned} \frac{2 x^{4} - 2}{x^{2} + 1} \end{aligned}$ .
::回顾矩形区域是宽度的长度倍数。要找到宽度, 我们可以将区域除以宽度。所以我们要寻找 2x4 - 2x2+1 。

If we factor the numerator and the denominator, we get:
::如果我们乘以分子和分母,我们就会得到:

$\begin{aligned} \frac{2 x^{4} - 2}{x^{2} + 1} \\ \frac{2 (x^{4} - 1)}{x^{2} + 2} \\ \frac{2 (x^{2} + 1) (x^{2} - 1)}{x^{2} + 1} \\ 2 (x^{2} - 1) = 2 x^{2} - 2 \end{aligned}$

::2x4-2x2+12(x4-1-1)x2+22(x2+1)(x2-1)x2+12(x2-1)=2x2-2

Therefore, the length of the rectangle is $\begin{aligned} 2 (x^{2} - 1) = 2 x^{2} - 2 \end{aligned}$ .
::因此,矩形长度为2(x2-1)=2x2-2。

If possible, simplify the following rational functions.
::如有可能,简化下列合理职能。

Example 2
::例2

$\begin{aligned} \frac{3 x^{2} - x}{3 x^{2}} \end{aligned}$
::3x2 - x3x2

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

Example 3
::例3

$\begin{aligned} \frac{x^{2} + 6 x + 8}{x^{2} + 6 x + 9} \end{aligned}$
::x2+6x+8x2+6x+9

$\begin{aligned} \frac{x^{2} + 6 x + 8}{x^{2} + 6 x + 9} = \frac{(x + 4) (x + 2)}{(x + 3) (x + 3)} \end{aligned}$ There are no common factors, so this is reduced.
::x2+6x+8x2+6x+9=(x+4)(x+2(x+2)(x+3)(x+3))没有共同因素,因此会减少。

Example 4
::例4

$\begin{aligned} \frac{2 x^{2} + x - 10}{6 x^{2} + 17 x + 5} \end{aligned}$
::2x2+x-106x2+17x+5

$\begin{aligned} \frac{2 x^{2} + x - 10}{6 x^{2} + 17 x + 5} = \frac{(2 x + 5) (x - 2)}{(2 x + 5) (3 x + 1)} = \frac{x - 2}{3 x + 1} \end{aligned}$
::2x2+x-106x2+17x+5=(2x+5)(x-2)(2x+5)(3x+1)=x-23x+1

Example 5
::例5

$\begin{aligned} \frac{x^{3} - 4 x}{x^{5} + 4 x^{3} - 32 x} \end{aligned}$
::x3-4x25+4x3-3-3-22xxx

In this problem, the denominator will factor like a quadratic once an $\begin{aligned} x \end{aligned}$ is pulled out of each term .
::在此问题上,当一个x从每个术语中拔出时,分母就会像二次方位数一样考虑。

$\begin{aligned} \frac{x^{3} - 4 x}{x^{5} + 4 x^{3} - 32 x} = \frac{x (x^{2} - 4)}{x (x^{4} + 4 x^{2} - 32)} = \frac{x (x - 2) (x + 2)}{x (x^{2} - 4) (x^{2} + 8)} = \frac{x (x - 2) (x + 2)}{x (x - 2) (x + 2) (x^{2} + 8)} = \frac{1}{x^{2} + 8} \end{aligned}$
::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

Review
::回顾

Does $\begin{aligned} \frac{x - 2}{x - 6} \end{aligned}$ simplify to $\begin{aligned} \frac{1}{3} \end{aligned}$ ? Explain why or why not.
::x-2x-6 是否简化为 13 ? 解释原因或原因。

Does $\begin{aligned} \frac{5 x}{10 x} \end{aligned}$ simplify to $\begin{aligned} \frac{1}{2} \end{aligned}$ ? Explain why or why not.
::5x10x是否简化为12?解释原因或原因。

In your own words, explain the difference between the previous two expressions and why one simplifies and one does not.
::用你自己的话来说,请解释前两个表达式的区别,以及一个表达式简化而一个不简化的原因。

Simplify the following rational expressions.
::简化以下理性表达式。

$\begin{aligned} \frac{4 x^{3}}{2 x^{2} + 3 x} \end{aligned}$
::432x2+3x

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

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

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

$\begin{aligned} \frac{8 x^{2} - 60 x - 32}{- 4 x^{2} + 26 x + 48} \end{aligned}$
::8x2-60x-32-44x2+26x+48

$\begin{aligned} \frac{6 x^{3} - 24 x^{2} + 30 x - 120}{9 x^{4} + 36 x^{2} - 45} \end{aligned}$
::6x3-24x2+30x-1209x4+36x2-45

$\begin{aligned} \frac{6 x^{2} + 5 x - 4}{6 x^{2} - x - 1} \end{aligned}$
::6x2+5x-46x2-x-1

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

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

$\begin{aligned} \frac{x^{5} - 3 x^{3} - 4 x}{x^{4} + 2 x^{3} + x^{2} + 2 x} \end{aligned}$
::x5 - 3x3 - 4x4+2x3+x2+2x

$\begin{aligned} \frac{- 3 x^{2} + 25 x - 8}{x^{3} - 8 x^{2} + x - 8} \end{aligned}$
::- 3x2+25x-8x3-8x2+x-8

$\begin{aligned} \frac{- x^{3} + 3 x^{2} + 13 x - 15}{- 2 x^{3} + 7 x^{2} + 20 x - 25} \end{aligned}$
::-x3+3x2+13x-15-2x3+7x2+20x-25

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

简化逻辑表达式

Rational Expressions ::理性表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Rational Expressions
::理性表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)