Course: 12.8 理性表达式的乘法 | The Way To Learn

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逻辑表达式乘法
Multiplication of Rational Expressions
::逻辑表达式乘法

The rules for multiplying and dividing rational expressions are the same as the rules for multiplying and dividing rational numbers, so let’s start by reviewing multiplication and division of fractions. When we multiply two fractions we multiply the numerators and denominators separately:
::计算和分解理性表达的规则与计算和分解合理数字的规则相同,因此,让我们首先审查乘法和分数的分法。当我们乘法为两个分数时,我们分别乘乘数和分母:

$\begin{aligned} \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \end{aligned}$

::abcd=accd =accd =accd =accd =accd =accd =accd =accd

Multiplying Rational Expressions Involving Monomials
::涉及单项的乘数逻辑表达式

1. Multiply the following: $\begin{aligned} \frac{a}{16 b^{8}} \cdot \frac{4 b^{3}}{5 a^{2}} \end{aligned}$ .
::1. 乘以: a16b8_4b35a2。

Cancel common factors from the numerator and denominator. The common factors are 4, $\begin{aligned} a \end{aligned}$ , and $\begin{aligned} b^{3} \end{aligned}$ . Canceling them out leaves $\begin{aligned} \frac{1}{4 b^{5}} \cdot \frac{1}{5 a} = \frac{1}{20 a b^{5}} \end{aligned}$ .
::分子和分母的取消系数。共同系数为 4,a和b3。取消系数14b515a=120ab5。

2. Multiply $\begin{aligned} 9 x^{2} \cdot \frac{4 y^{2}}{21 x^{4}} \end{aligned}$ .
::2. 乘以 9x2+4y221x4。

Rewrite the problem as a product of two fractions: $\begin{aligned} \frac{9 x^{2}}{1} \cdot \frac{4 y^{2}}{21 x^{4}} \end{aligned}$ Then cancel common factors from the numerator and denominator.
::将问题重写为两个分数的产物: 9x214y2221x4, 然后从分子和分母中取消共同系数。

The common factors are 3 and $\begin{aligned} x^{2} \end{aligned}$ . Canceling them out leaves $\begin{aligned} \frac{3}{1} \cdot \frac{4 y^{2}}{7 x^{2}} = \frac{12 y^{2}}{7 x^{2}} \end{aligned}$ .
::常见因素是 3 和 x2 。取消它们 314y27x2 = 12y27x2 。

Multiplying Rational Expressions Involving Polynomials
::涉及聚合体的乘数逻辑表达式

When multiplying rational expressions involving , first we need to factor all polynomial expressions as much as we can. Then we follow the same procedure as before.
::当乘以包含的理性表达式时, 首先我们需要尽可能多地考虑所有多面性表达式。然后我们遵循与之前相同的程序。

Multiply $\begin{aligned} \frac{4 x + 12}{3 x^{2}} \cdot \frac{x}{x^{2} - 9} \end{aligned}$ .
::乘以 4x+123x2xxxx22-9。

Factor all polynomial expressions as much as possible: $\begin{aligned} \frac{4 (x + 3)}{3 x^{2}} \cdot \frac{x}{(x + 3) (x - 3)} \end{aligned}$
::尽可能将所有多多边表达式的系数数数数: 4 (x+3)3x2xx(x+3)(x- 3)

The common factors are $\begin{aligned} x \end{aligned}$ and $\begin{aligned} (x + 3) \end{aligned}$ . Canceling them leaves $\begin{aligned} \frac{4}{3 x} \cdot \frac{1}{(x - 3)} = \frac{4}{3 x (x - 3)} = \frac{4}{3 x^{2} - 9 x} \end{aligned}$ .
::常见因素是 x 和 (x+3) 。取消它们的叶子为 43x1(x-3) = 43x(x-3) = 43x2- 9x 。

Multiplying a Rational Expression by a Polynomial
::以多面性乘以逻辑表达式

When we multiply a rational expression by a whole number or a polynomial, we can write the whole number (or polynomial) as a fraction with denominator equal to one. We then proceed the same way as in the previous examples.
::当我们将一个理性表达方式乘以整数或多数值时,我们可以将整个数字(或多数值)写成一个分数,其分母等于一个分母。然后我们按前例一样的方式进行。

Multiply $\begin{aligned} \frac{3 x + 18}{4 x^{2} + 19 x - 5} \cdot (x^{2} + 3 x - 10) \end{aligned}$ .
::乘以 3x+184x2+19x-5(x2+3x-10)。

Rewrite the expression as a product of fractions: $\begin{aligned} \frac{3 x + 18}{4 x^{2} + 19 x - 5} \cdot \frac{x^{2} + 3 x - 10}{1} \end{aligned}$
::将表达式重写为分数的产物: 3x+184x2+19x- 5xx2+3x-101

Factor polynomials: $\begin{aligned} \frac{3 (x + 6)}{(x + 5) (4 x - 1)} \cdot \frac{(x - 2) (x + 5)}{1} \end{aligned}$
::系数多元数: 3(x+6)(x+5)(4x-1)(x-2)(x+5)1

The common factor is $\begin{aligned} (x + 5) \end{aligned}$ . Canceling it leaves $\begin{aligned} \frac{3 (x + 6)}{(4 x - 1)} \cdot \frac{(x - 2)}{1} = \frac{3 (x + 6) (x - 2)}{(4 x - 1)} = \frac{3 x^{2} + 12 x - 36}{4 x - 1} \end{aligned}$
::共同系数是 (x+5) 。取消时留下的值为 3( x+6)(4x- 1) = (x-2) 1= 3 (x+6)(x-2)(4x- 1) = 3x2+12x-364x-1 。

Example
::示例示例示例示例

Example 1
::例1

Multiply $\begin{aligned} \frac{12 x^{2} - x - 6}{x^{2} - 1} \cdot \frac{x^{2} + 7 x + 6}{4 x^{2} - 27 x + 18} \end{aligned}$ .
::乘以 12x2-x-6x2-1x2+7x+64x2-27x+18。

Factor polynomials: $\begin{aligned} \frac{(3 x + 2) (4 x - 3)}{(x + 1) (x - 1)} \cdot \frac{(x + 1) (x + 6)}{(4 x - 3) (x - 6)} \end{aligned}$ .
::系数多元数: (3x+2)(4x-3)(x+1)(x-1)(x-1)(x+1)(x+6)(4x-3)(x-6)

The common factors are $\begin{aligned} (x + 1) \end{aligned}$ and $\begin{aligned} (4 x - 3) \end{aligned}$ . Canceling them leaves $\begin{aligned} \frac{(3 x + 2)}{(x - 1)} \cdot \frac{(x + 6)}{(x - 6)} = \frac{(3 x + 2) (x + 6)}{(x - 1) (x - 6)} = \frac{3 x^{2} + 20 x + 12}{x^{2} - 7 x + 6} \end{aligned}$
::常见因素是(x+1)和(4x-3)。取消树叶(3x+2)(x-1)(x+6)(x-6)=(3x+2)(x+2)(x+6)(x+6)(x-1)(x-6)(x-6)(x-1)(x-6)=3x2+20x+12x2-7x+6)

Review
::回顾

Multiply the following rational expressions and reduce the answer to lowest terms.
::乘以下列合理表达方式,将答案降低到最低值。

$\begin{aligned} \frac{x^{3}}{2 y^{3}} \cdot \frac{2 y^{2}}{x} \end{aligned}$
::x32y32y2x

$\begin{aligned} \frac{2 x}{y^{2}} \cdot \frac{4 y}{5 x} \end{aligned}$
::2x24y5x

$\begin{aligned} 2 x y \cdot \frac{2 y^{2}}{x^{3}} \end{aligned}$
::2x=2x2y2x3

$\begin{aligned} \frac{4 y^{2} - 1}{y^{2} - 9} \cdot \frac{y - 3}{2 y - 1} \end{aligned}$
::4y2-一y2-九-三十二-1

$\begin{aligned} \frac{6 a b}{a^{2}} \cdot \frac{a^{3} b}{3 b^{2}} \end{aligned}$
::6aba2a3b3b2

$\begin{aligned} \frac{33 a^{2}}{- 5} \cdot \frac{20}{11 a^{3}} \end{aligned}$
::2011年5月3日a3

$\begin{aligned} \frac{2 x^{2} + 2 x - 24}{x^{2} + 3 x} \cdot \frac{x^{2} + x - 6}{x + 4} \end{aligned}$
::2x2+2x-24x2+3xxx2+x-6x+4

$\begin{aligned} \frac{x}{x - 5} \cdot \frac{x^{2} - 8 x + 15}{x^{2} - 3 x} \end{aligned}$
::xxx-5xx2-8x+15x2-3x

$\begin{aligned} \frac{5 x^{2} + 16 x + 3}{36 x^{2} - 25} \cdot (6 x^{2} + 5 x) \end{aligned}$
::5x2+16x+336x2-25}(6x2+5x)

$\begin{aligned} \frac{x^{2} + 7 x + 10}{x^{2} - 9} \cdot \frac{x^{2} - 3 x}{3 x^{2} + 4 x - 4} \end{aligned}$
::x2+7x+10x2-9x2-3x3x2+4x-4

$\begin{aligned} \frac{x^{2} + 8 x + 16}{7 x^{2} + 9 x + 2} \cdot \frac{7 x + 2}{x^{2} + 4 x} \end{aligned}$
::x2+8x+167x2+9x+2=7x+2x2x2+4x

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。

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