Course: 9.10 添加和减减与类似输入器一样的理性表达式

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添加和减减有类似引号的理性表达式
In triangle ABC, side AB is $\begin{align*}\frac{x^2+5}{x^2 + 3x + 2}\end{align*}$ units long. Side AC is $\begin{align*}\frac{3x^2 - 3x}{x^2 + 3x + 2}\end{align*}$ units long. How much longer is side AC than side AB?
::在三角ABC, 侧AB是 x2+5x2+3x+2 单位。侧AC 是 3x2-3x2+3x2 单位。侧AC 比侧AB 还要长多久?

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

Recall, that when you add or subtract fractions, the denominators must be the same. The same is true of adding and subtracting rational expressions. The denominators must be the same expression and then you can add or subtract the numerators.
::回想,当您添加或减减分数时,分母必须是相同的。增减理性表达式的情况相同。分母必须是相同的表达式,然后您可以添加或减减分子。

Let's add or subtract the following rational expressions.
::我们加或减以下合理表达式。

Add $\begin{align*}\frac{x}{x-6} + \frac{7}{x-6}\end{align*}$ .
::添加xx-6+7x-6。

In this concept, the denominators will always be the same. Therefore , all you will need to do is add the numerators and simplify if needed.
::在这个概念中,分母总是相同的。因此,你需要做的就是添加分子,必要时简化。

$\begin{align*}\frac{x}{x-6} + \frac{7}{x-6} = \frac{x+7}{x-6}\end{align*}$
::xx-6+7x-6=x+7x-6

Subtract $\begin{align*}\frac{x^2-4}{x-3} - \frac{2x-1}{x-3}\end{align*}$ .
::减号 x2 - 4x - 3 - 2x - 1x - 3 。

You need to be a little more careful with subtraction . The entire expression in the second numerator is being subtracted. Think of the minus sign like distributing -1 to that numerator.
::您需要稍加小心减法。第二个分子中的整个表达式正在被减。将减号像分配 - 1 给该分子一样, 想一想减号。

$\begin{aligned} \frac{x^{2} - 4}{x - 3} - \frac{2 x - 1}{x - 3} & = \frac{x^{2} - 4 - (2 x + 1)}{x - 3} \\ = \frac{x^{2} - 4 - 2 x - 1}{x - 3} \\ = \frac{x^{2} - 2 x - 3}{x - 3} \end{aligned}$
$\begin{align*}\frac{x^2-4}{x-3} - \frac{2x-1}{x-3} &= \frac{x^2-4-(2x+1)}{x-3} \\ &= \frac{x^2-4-2x-1}{x-3} \\ &= \frac{x^2-2x-3}{x-3}\end{align*}$
::x2 - 4x - 3 - 2x - 2x - 1x - 3=x2 - 4 - (2x+1)x - 3x=x2 - 4 - 2x - 1x - 3=x2 - 2x - 3x - 3

At this point, factor the numerator if possible.
::在此点,如果可能,请乘以分子。

$\begin{align*}\frac{x^2-2x-3}{x-3} = \frac{\cancel{(x-3)}(x+1)}{\cancel{x-3}} = x+1\end{align*}$
::x2-2x-3x-3=(x-3)(x+1)x-3=x1)

Add $\begin{align*}\frac{x+7}{2x^2+14x+20} + \frac{x+1}{2x^2+14x+20}\end{align*}$ .
::添加 x+72x2+14x+20+x+12x2+14x+20。

Add the numerators and simplify the denominator.
::添加分子并简化分母。

$\begin{aligned} \frac{x + 7}{2 x^{2} + 14 x + 20} + \frac{x + 1}{2 x^{2} + 14 x + 20} & = \frac{2 x + 8}{2 x^{2} + 14 x + 20} \\ = \frac{2 (x + 4)}{2 (x + 5) (x + 2)} \\ = \frac{(x + 4)}{(x + 5) (x + 2)} \end{aligned}$
$\begin{align*}\frac{x+7}{2x^2+14x+20} + \frac{x+1}{2x^2+14x+20} &= \frac{2x+8}{2x^2+14x+20} \\ &= \frac{\cancel{2}(x+4)}{\cancel{2}(x+5)(x+2)} \\ &= \frac{(x+4)}{(x+5)(x+2)}\end{align*}$
::x+72x2+14x+20+x+12x2+14x+20=2x82x2+14x+20=2(x+4)2(x+5)(x+2)=(x+4)(x+5)(x+2)=(x+4)(x+5)(x+2)

Examples
::实例

Example 1
::例1

Earlier, you were asked to find how much longer side AC is compared to side AB.
::早些时候,你被要求去寻找 AC 和AB 之间的比对距离有多长。

We need to subtract the length of side AB from the length of side AC.
::我们需要将AB方的长度从AC方的长度中减去。

$\begin{aligned} \frac{3 x^{2} - 3 x}{x^{2} + 3 x + 2} - \frac{x^{2} + 5}{x^{2} + 3 x + 2} \\ \frac{3 x^{2} - 3 x - (x^{2} + 5)}{x^{2} + 3 x + 2} \\ \frac{3 x^{2} - 3 x - x^{2} - 5}{x^{2} + 3 x + 2} \\ \frac{2 x^{2} - 3 x - 5}{x^{2} + 3 x + 2} \end{aligned}$
$\begin{align*}\frac{3x^2 - 3x}{x^2 + 3x + 2} - \frac{x^2+5}{x^2 + 3x + 2}\\ \frac{3x^2 - 3x - (x^2+5)}{x^2 + 3x + 2}\\ \frac{3x^2 - 3x - x^2 - 5}{x^2 + 3x + 2}\\ \frac{2x^2 - 3x - 5}{x^2 + 3x + 2}\end{align*}$
::3x2 - 3x2+3x3x2+3x2 - x2+5x2+3x3x23x2 - 3x(x2+5+5)xx2+3x3x23x2+23x2 - 3x - 5x2 - 5x2+3x2+3x2x3x5x2+22x2 -3x2 -5x2+3x2+3x2

Now we need to factor the numerator and the denominator.
::现在我们需要考虑分子和分母。

$\begin{align*}\frac{2x^2 - 3x - 5}{x^2 + 3x + 2} = \frac{(2x-5)(x+1)}{(x+2)(x+1)}\end{align*}$
::2x2-3x-5x2+3x+2=(2x-5)(x+1)(x+2)(x+1)(x+1)

The $\begin{align*}(x + 1)\end{align*}$ in the numerator and the denominator cancel out and we are left with $\begin{align*}\frac{2x-5}{x+2}\end{align*}$ . Therefore, side AC is $\begin{align*}\frac{2x-5}{x+2}\end{align*}$ units longer than side AB.
::分子中( x+1) 和分母取消后, 我们只剩下 2x-5x+2 。因此, 侧AC 是 2x-5x+2 单位, 比侧AB 更长。

Example 2
::例2

Subtract $\begin{align*}\frac{3}{x^2-9} - \frac{x+7}{x^2-9}\end{align*}$ .
::减号 3x2-9-x+7x2-9。

$\begin{align*}\frac{3}{x^2-9} - \frac{x+7}{x^2-9} = \frac{3-(x+7)}{x^2-9} = \frac{3-x-7}{x^2-9} = \frac{-x-4}{x^2-9}\end{align*}$
::3x2-9-x7x2-9=3-(x+7)x2-9=3-(x+7)x2-9=3-x-7x2-9*x-4x2-9

We did not bother to factor the denominator because we know that the factors of -9 are 3 and -3 and will not cancel with $\begin{align*}-x-4\end{align*}$ .
::我们没有考虑分母,因为我们知道-9因素是3和3,不会取消-x-4。

Example 3
::例3

Add $\begin{align*}\frac{5x-6}{2x+3} + \frac{x-12}{2x+3}\end{align*}$ .
::添加 5x-62x+3+x-122x+3。

$\begin{align*}\frac{5x-6}{2x+3} + \frac{x-12}{2x+3} = \frac{6x-18}{2x+3} = \frac{6(x-3)}{2x+3}\end{align*}$
::5x-62x+3+3+x-122x+3=6x-182x+3=6(x-3-3)2x+3

Example 4
::例4

Subtract $\begin{align*}\frac{x^2+2}{4x^2-4x-3} - \frac{x^2-2x+1}{4x^2-4x-3}\end{align*}$ .
::减号 x2+24x2 - 4x - 3 - x2 - 2x+14x2 - 4x - 3 。

$\begin{aligned} \frac{x^{2} + 2}{4 x^{2} - 4 x - 3} - \frac{x^{2} - 2 x + 1}{4 x^{2} - 4 x - 3} & = \frac{x^{2} + 2 - (x^{2} - 2 x + 1)}{4 x^{2} - 4 x - 3} \\ = \frac{x^{2} + 2 - x^{2} + 2 x - 1}{4 x^{2} - 4 x - 3} \\ = \frac{2 x + 1}{4 x^{2} - 4 x - 3} \end{aligned}$
$\begin{align*}\frac{x^2+2}{4x^2-4x-3} - \frac{x^2-2x+1}{4x^2-4x-3} &= \frac{x^2+2-(x^2-2x+1)}{4x^2-4x-3} \\ &= \frac{x^2+2-x^2+2x-1}{4x^2-4x-3} \\ &= \frac{2x+1}{4x^2-4x-3}\end{align*}$
::x2 - 24x2 - 4x2 - 4x - 3 - 2 - 2x+14x2 - 4x - 3=x2+2 (x2 - 2x+1x) - 4x2 - 4x2 - 4x - 3=x2 - 2x2 - 2x2 - 2x2 - 4x - 4x3=2x142 - 4x2 - 4x3

At this point, we will factor the denominator to see if any factors cancel with the numerator.
::在这一点上,我们将考虑分母,以确定是否有任何因素与分子抵消。

$\begin{align*}\frac{2x+1}{4x^2-4x-3} = \frac{\cancel{2x+1}}{\cancel{(2x+1)}(2x-3)} = \frac{1}{2x-3}\end{align*}$
::2x+14x2-4x-3=2x1(2x+1)(2x-3)=12x-3

Review
::回顾

Explain how you add fractions. Assume your audience knows nothing about math.
::解释你如何添加分数。假设你的观众对数学一无所知。

Explain why $\begin{align*}\frac{2}{3} + \frac{4}{5} \neq \frac{3}{4}\end{align*}$ .
::解释为什么23+45+34。

Add or subtract the following rational expressions.
::添加或减去以下理性表达式。

$\begin{align*}\frac{2}{x} + \frac{5}{x}\end{align*}$
::2x+5x

$\begin{align*}\frac{5}{2x} + \frac{7}{2x}\end{align*}$
::52x52+72x

$\begin{align*}\frac{6}{5x} + \frac{3-2x}{5x}\end{align*}$
::65x+3-2x5x

$\begin{align*}\frac{3}{x} + \frac{x+1}{x}\end{align*}$
::3x+x+1x 3x+x+1x

$\begin{align*}\frac{5}{x+1} + \frac{x-4}{x+1}\end{align*}$
::5x+1+x-4x+1

$\begin{align*}\frac{x+15}{x-2} - \frac{10}{x-2}\end{align*}$
::x+15x-2-10x-2

$\begin{align*}\frac{4x-3}{x+3} + \frac{15}{x+3}\end{align*}$
::4 - 3x+3+15x+3

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

$\begin{align*}\frac{5x+3}{x^2-4} - \frac{2x+9}{x^2-4}\end{align*}$
::5x+3x2 - 4 - 2x+9x2 - 4

$\begin{align*}\frac{3x^2+x}{x^3-8} + \frac{4}{x^3-8} - \frac{2x^2-x}{x^3-8}\end{align*}$
::3x2+x23-8+4x3-8-8-2x2-x23-8

$\begin{align*}\frac{4x+3}{x^2+1} - \frac{x+2}{x^2+1} + \frac{1-x}{x^2+1}\end{align*}$
::4x+3x2+1 - x2x2+1+1 -x2+1+1 -x2+1

$\begin{align*}\frac{18x^2-7x+2}{8x^3+4x^2-18x-9} - \frac{3x^2+13x-4}{8x^3+4x^2-18x-9} + \frac{5x^2-13}{8x^3+4x^2-18x-9}\end{align*}$
::18x2-7x+28x3+28x3+4x3+4x2-18x2-8x9-3x2+13x-48x3+4x2-18x2-18x9+5x2-138x3+4x2-18x9

$\begin{align*}\frac{2x^2+3x}{x^3+2x^2-16x-32} + \frac{5x^2-13}{x^3+2x^2-16x-32} - \frac{4x^2+9x+11}{x^3+2x^2-16x-32}\end{align*}$
::2x2+3x23+2x2 - 16x - 32+5x2 - 13x3+2x2 - 16x - 32 - 4x2+9x+11x3+2x2 - 16x32

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

添加和减减有类似引号的理性表达式

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

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)

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

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)