课程： 6.5 乘数和分裂有理表达式-interactive

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乘和 divide 逻辑表达式

Lesson Objectives
::经验教训目标

Understand that rational expressions form a system analogous to the rational numbers, closed under multiplication and division by a nonzero rational expression .
::理解理性表达方式形成一个类似于理性数字的体系,在乘法下闭合,以非零理性表达方式隔开。

Multiply and divide rational expressions.
::乘法和分裂合理表达方式。

Introduction: Early Retirement
::导言:提前退休

It is important to have a strong understanding of how retirement plans work. With careful planning and an understanding of the proper math, you can set yourself up to retire with enough money saved away to live comfortably.
::重要的是要深入地了解退休计划是如何运作的。仔细地规划并理解正确的数学,你就可以用足够的储蓄来安稳地生活。

A retirement fund uses compound interest to grow money saved in the account. However, you don't put a set amount into the account and let it grow. Instead, you add a set amount each pay period to grow your account while the interest grows. The formula for calculating the amount in the account as it grows with compound interest is shown below:
::退休基金使用复利来增加账户中储蓄的金额。但是, 您不会在账户中设置一个固定金额, 并让账户增长。相反, 您会在每个支付期添加一个固定金额, 以便在利息增长的同时增加您的账户。计算账户中金额的公式与复利增长如下:

\begin{aligned} A = P {(1 + \frac{r}{n})}^{r t} + \frac{c ({(1 + \frac{r}{n})}^{r t} - 1)}{\frac{r}{n}} \end{aligned}

::A=P(1+rn)rt+c(1+rn)rt-1nn

$\begin{aligned} A \end{aligned}$ is the amount in the retirement fund
::A 是退休基金中的数额。

$\begin{aligned} P \end{aligned}$ is the principle
::P是原则

$\begin{aligned} r \end{aligned}$ is the interest rate
::r 是利率

$\begin{aligned} n \end{aligned}$ is the number of times the fund is compounded in a year
::n 是基金一年中复数的倍数。

$\begin{aligned} t \end{aligned}$ is the number of years
::t 是年数

$\begin{aligned} c \end{aligned}$ is the contribution made each pay period
::c 是指每个支付期的缴款额。

The formula above starts as the basic compound interest formula. This part of the equation calculates the value upon retirement of the starting amount in the account. The right side of the expression will produce the value of the contributions, including interest, over time.
::以上公式开始为基本复合利息公式。方程式的这一部分计算账户中起始金额退出时的数值。表达式的右侧将产生缴费值, 包括利息, 并随着时间推移产生。

Clearly, as you venture further into math, the formulas get more complicated. Before you can understand formulas like the one above, you will need to challenge your understanding of fraction operations .
::显然,随着你进一步投入数学,公式会变得更加复杂。在你能够理解上述公式之前,你需要挑战你对分数操作的理解。

Activity 1: Multiplying Single Term Fractions
::活动1:乘数单期分数

This activity will expand basic knowledge of multiplying numerical fractions into multiplying more complex algebraic expressions. To multiply fractions, multiply the numerator by the numerator and the denominator by the denominator.
::此活动将扩展将数字分数乘以更复杂的代数表达式的基本知识。要将分数乘以分数, 则要将分子乘以分子, 分母乘以分母。

\begin{aligned} \frac{3}{5} \cdot \frac{7}{4} = \frac{21}{20} \end{aligned}

Additionally, remember that you can cross cancel common factors in the numerator and the denominator of a fraction.
::此外,请记住您可以交叉勾销分子和分数分母中的常见系数。

\begin{aligned} \frac{9}{10} \cdot \frac{5}{12} = \frac{3 \cdot 3}{2 \cdot 5} \cdot \frac{5}{2 \cdot 2 \cdot 3} = \frac{3}{8} \end{aligned}

More complicated algebraic expressions can have polynomial factors. The expression $\begin{aligned} x^{3} - 5 x^{2} + 6 x = x (x - 2) (x - 3) \end{aligned}$ has both binomial factors and monomial factors. You can multiply and cancel these algebraic fractions the same way that you can with the numerical fractions above.
::更复杂的代数表达式可能有多数值因子。表达式 x3- 5x2+6x=x(x- 2)(x-3) 既有二进制因子,也有单数因子。您可以与上面的数分数一样,乘以并取消这些代数分数。

Example
::示例示例示例示例

Multiply the following rational expressions, and s implify the result.
::乘以下列合理表达式,并简化结果。

\begin{aligned} \frac{4 x^{2} y^{5} z}{6 x y z^{6}} \cdot \frac{15 y^{4}}{35 x^{4}} \end{aligned}

::4x2y5z6xyz615y435x4

These rational expressions are monomials with multiple variables. R e call the laws of exponents : add the exponents when multiplying and subtract the exponents when dividing. One way to solve this type of problem is to multiply the two fractions together first and then reduce the product . Another method is to cancel any common factors across fractions before multiplying:
::这些理性表达式是带有多个变量的单项表达式。回顾引言法则 : 乘以引言法则, 乘以引言法则, 乘以引言法则, 乘以引言法则。解决这类问题的一种方法是先将两个分数相乘, 然后再减少产品。另一种方法是在乘以前取消各分数之间的任何共同因素 :

\begin{aligned} \frac{7 x^{2} y^{3} z}{6 x y z^{2}} \cdot \frac{15 y^{4}}{28 x} = \frac{7 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot z}{2 \cdot 3 \cdot x \cdot y \cdot z \cdot z} \cdot \frac{3 \cdot 5 \cdot y \cdot y \cdot y \cdot y}{2 \cdot 2 \cdot 7 \cdot x} = \frac{5 y^{6}}{8 z} \end{aligned}

::7x2y3z6xyz2

Answer: $\begin{aligned} \frac{5 y^{6}}{8 z} \end{aligned}$
::答复:5y68z

INTERACTIVE

Simplify Variable Expressions

When dividing like terms, you can use the commutative and associative properties of division to group and simplify the coefficients and variables separately.
::当像用词一样的分隔时,您可以使用除法的通和和关联属性来分组和分别简化系数和变量。

Slide the colored points at the left of each expression to change the colored values.
::将每个表达式左侧的彩色点幻灯片,以更改彩色值。

See how changing the coefficients and powers of variables changes how the expressions are simplified.
::看看变数的系数和功率的变化如何改变表达式的简化。

roducts-and-quotients-of-rational-expressions" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5f20455da638fb2172e38ffc&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:

roducts-and-quotients-of-rational-expressions&mode=lite" test-id="5f20455da638fb2172e38ffc">

Discussion Question : Does it matter whether you multiply the fractions before reducing or cross cancel common factors before multiplying ? Is one process going to be more efficient than the other?
::讨论问题:您在减少或交叉取消共同因素之前先乘数分数,然后再乘数是否重要?一个过程是否比另一个过程更有效率?

Activity 2: Multiplying Rational Expressions
::活动2:乘数逻辑表达式

To multiply rational expressions, use the following steps:
::要乘以理性表达,请使用以下步骤:

Factor the numerator and denominator of each fraction.
::乘以每个分数的分子和分母。

Cross out any common factors and multiply any remaining factors.
::排除任何共同因素,乘以任何剩余因素。

Example
::示例示例示例示例

\begin{aligned} f (x) = \frac{x^{2} + 6 x - 7}{x^{2} - 36} \end{aligned}

and

\begin{aligned} g (x) = \frac{x^{2} - 2 x - 24}{2 x^{2} + 8 x - 42}, \end{aligned}

find

\begin{aligned} (f \cdot g) (x) . \end{aligned}

1. Factor the numerator and denominator:
::1. 乘以分子和分母:

\begin{aligned} \frac{x^{2} + 6 x - 7}{x^{2} - 36} \cdot \frac{x^{2} - 2 x - 24}{2 x^{2} + 8 x - 42} = \frac{(x + 7) (x - 1)}{(x - 6) (x + 6)} \cdot \frac{(x - 6) (x + 4)}{2 (x + 7) (x - 3)} \end{aligned}

::x2+6x-7x2 - 36x2 - 2x - 242x2+8x-42=(x+7x-1)(x-6)(x+6)(x+6)(x-6)(x-6)(x-6)(x+4)(x+4)(x+7)(x-3)

2. Cross out any common factors and multiply any remaining factors.
::2. 排除任何共同因素,乘以任何剩余因素。

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

x+7)(x-1)(x-6)(x-6)(x-6)(x-6)(x-6)(x+4)(x-6)(x-6)(x+4)(2)(x+7)(x-3)=(x-1)(x+6)(x+6)(x+4)(2x-3)

Answer: $\begin{aligned} \frac{(x - 1) (x + 4)}{2 (x + 6) (x - 3)} \end{aligned}$
::答复x-1)(x+4)(2x+6)(x-3)

You could multiply the numerator and denominator out, but it is not necessary.
::您可以乘以分子和分母, 但没有必要。

INTERACTIVE

Multiplying Rational Expressions

Click through the steps to multiply a rational expression by a polynomial.
::单击各个步骤,将一个理性表达式乘以一个多式表达式。

Remember these three steps for multiplying all rational expressions.
::记住这三步所有理性表达方式的乘法

roducts-and-quotients-of-rational-expressions" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5f204f6b8b5b7723b7db66d1&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:

roducts-and-quotients-of-rational-expressions&mode=lite" test-id="5f204f6b8b5b7723b7db66d1">

Discussion Question : A polynomial multiplied by a polynomial will always equal a polynomial. C an you use this to prove that a rational expression multiplied by a rational expression will always result in a rational expression?
::讨论问题 : 多式乘以多式将永远等于多式。您能否用这个来证明一个理性表达方式乘以一个理性表达方式将永远导致一个理性表达方式?

The interactive below will help you practice multiplying rational expressions.
::下面的交互效果会帮助您练习乘以理性表达方式。

Extension: Multiplying Rational Expressions Video
::扩展名: 视频

This video demonstrates how to multiply rational expressions.
::这段影片展示了如何乘以理性表达方式。

Activity 3: Dividing Single Term Fractions
::活动3:单期分部分

The formula in the introduction involved dividing rational expressions. Revisit dividing numerical fractions b efore investigating more complex cases of division involving rational expressions.
::在调查涉及合理表达的更复杂的分裂案件之前,重新研究数字分数,然后调查涉及合理表达的较复杂的分裂案件。

INTERACTIVE

Dividing Single Term Fractions

Move the red point to see how rational numbers behave when multiplied or divided.
::移动红点以查看乘或除时合理数字的运行方式。

To divide fractions, take the second fraction, flip it, and change the operator to multiplication. Multiplying by the reciprocal applies to fractions featuring rational expressions as well. Luckily, you already know how to multiply fractions with rational expressions.
::要分割分数, 取第二个分数, 翻转它, 将运算符转换为乘法。乘以对等法也可以适用于带有理性表达方式的分数。幸运的是, 您已经知道如何以理性表达方式乘以分数。

Example
::示例示例示例示例

Divide the following rational expressions: $\begin{aligned} \frac{5 a^{3} b^{4}}{12 a b^{8}} \div \frac{15 b^{6}}{8 a^{6}} \end{aligned}$
::除以下列合理表达式: 5a3b412ab815b68a6

S tart by flipping the second fraction and changing the $\begin{aligned} \div \end{aligned}$ sign to multiplication.
::开始翻转第二个分数, 并更改 {} 符号为乘法。

\begin{aligned} \frac{5 a^{3} b^{4}}{12 a b^{5}} \div \frac{15 b}{8 a} = \frac{5 a^{3} b^{4}}{12 a b^{5}} \cdot \frac{8 a}{15 b} \end{aligned}

::5a3b412ab515b8a=5a3b412ab5_8a15b

From here, multiply the fractions as seen above.
::从这里乘以上文所见的分数。

\begin{aligned} \frac{5 a^{3} b^{4}}{12 a^{2}} \cdot \frac{8 a}{15 b^{5}} = \frac{5 \cdot a \cdot a \cdot a \cdot b \cdot b \cdot b \cdot b}{2 \cdot 2 \cdot 3 \cdot a \cdot a} \cdot \frac{2 \cdot 2 \cdot 2 \cdot a}{3 \cdot 5 \cdot b \cdot b \cdot b \cdot b \cdot b} = \frac{2 a^{2}}{45 b} \end{aligned}

::5a3b412 a2b412 a2b8a15b5=5aaaaaabbbb=2a245bb=2a245b

Answer: $\begin{aligned} \frac{2 a^{2}}{45 b} \end{aligned}$
::答复:2a245b

roducts-and-quotients-of-rational-expressions" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5f206e385db0c2a4a66c135a&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:

roducts-and-quotients-of-rational-expressions&mode=lite" test-id="5f206e385db0c2a4a66c135a">

Activity 4: Dividing Rational Expressions
::活动4:分裂的理性表达式

To divide more , you will need to add one step to the process of multiplying rational expressions.
::要进一步划分, 您需要为乘以理性表达的过程再加一步。

Change the operation to multiplication and write the reciprocal of the second fraction.
::将操作更改为乘法, 并写入第二分数的对等值。

Factor the numerator and the denominator of each fraction.
::乘以每个分数的分子和分母。

Cross out any common factors and multiply the remaining fractions.
::排出任何共同因素并乘以剩余的分数。

Example
::示例示例示例示例

Divide the following rational expressions: $\begin{aligned} \frac{x^{4} - 3 x^{2} - 4}{2 x^{2} + x - 10} \div \frac{x^{3} - 3 x^{2} + x - 3}{x - 2} \end{aligned}$
::除以下列合理表达式: x4- 3x2- 42x2+x- 10x3- 3x2+x- 3x2

1. Change the operation to multiplication and write the reciprocal of the second fraction.
::1. 将操作改为乘法,并写出第二分数的对等值。

\begin{aligned} \frac{x^{4} - 3 x^{2} - 4}{2 x^{2} + x - 10} \div \frac{x^{3} - 3 x^{2} + x - 3}{x - 2} & = \frac{x^{4} - 3 x^{2} - 4}{2 x^{2} + x - 10} \cdot \frac{x - 2}{x^{3} - 3 x^{2} + x - 3} \end{aligned}

::x4 - 3x2 - 3x2 - 42x2 - 4x2 - 4x2 - 10x2 - 3x3 - 3x2 - 3x2 - 3x2 - 3x2 - 42x2 - 3x2 - 10x2 - x3 - 3x2 - 3x2 - 3x3

2. Factor the numerator and denominator of each fraction.
::2. 乘以每一分数的分子和分母。

Flip the second fraction, change the $\begin{aligned} \div \end{aligned}$ sign to multiplication, and simplify .
::翻转第二个分数, 将 {} 符号更改为乘法, 并简化。

\begin{aligned} \frac{x^{4} - 3 x^{2} - 4}{2 x^{2} + x - 10} \cdot \frac{x - 2}{x^{3} - 3 x^{2} + x - 3} & = \frac{(x - 2) (x + 2) (x^{2} + 1)}{(2 x + 5) (x - 2)} \cdot \frac{x - 2}{(x^{2} + 1) (x - 3)} \end{aligned}

::x4 - 3x2 - 3x2 - 42x2+x - 10xx2 - 2x3 - 3x2 - 3x2=(x-2)(x2+2)(x2+2)(2x+1)(2x+5)(x-2)(x-2)xx - 2(x2+1)(x-3)

3. Cross out any common factors and multiply the remaining fractions.
::3. 排除任何共同因素,乘以剩余的分数。

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

x-2)(x+2)(x2+2)(x2+1)(2x+5)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x+1)(x-3))

Taking the product of the remaining expressions will give the answer.
::其余表达方式的产物将给出答案。

Answer: $\begin{aligned} \frac{(x - 2) (x + 2)}{(2 x - 5) (x - 3)} \end{aligned}$
::答复: (x-2)(x+2)(2x-5)(x-3)

INTERACTIVE

Dividing Rational Expressions

Click through the steps to divide rational expressions. Remember these four steps for dividing all rational expressions.
::单击用于分隔合理表达的阶梯。记住这四个阶梯来分隔所有合理表达的阶梯。

roducts-and-quotients-of-rational-expressions" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5f2071c0067f63a348e761c0&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:

roducts-and-quotients-of-rational-expressions&mode=lite" test-id="5f2071c0067f63a348e761c0">

Discussion Question : A polynomial divided by a polynomial does not always result in a polynomial. Does a rational expression divided by a rational expression always result in a rational expression? Prove your answer.
::讨论问题:由多面体分裂的多面体并不总能产生多面体。由理性表达法分裂的理性表达法是否总能产生理性表达法?证明您的回答。

This interactive will help you divide rational expressions.
::此互动会帮助您区分合理的表达方式。

Extension: Dividing Rational Expressions Video
::扩展名: Dividing 理性表达式视频

The video below demonstrates how to divide rational expressions.
::以下影片展示了如何划分合理表达方式。

Summary

Multiplying and dividing rational expressions is very similar to multiplying dividing fractions.
::乘法和分法合理表达方式与乘法分数非常相似。

To multiply rational expressions:
- Factor the numerator and denominator of each fraction.
  ::乘以每个分数的分子和分母。
- Cross out any common factors and multiply any remaining factors.
  ::排除任何共同因素,乘以任何剩余因素。
::乘以理性表达式 : 乘以每个分数的分子和分母。划出任何共同因素并乘以任何剩余因素。

To divide rational expressions:
- Change the operation to multiplication and write the reciprocal of the second fraction.
  ::将操作更改为乘法, 并写入第二分数的对等值。
- Factor the numerator and the denominator of each fraction.
  ::乘以每个分数的分子和分母。
- Cross out any common factors and multiply the remaining fractions.
  ::排出任何共同因素并乘以剩余的分数。
::区分合理表达式 : 将操作改为乘法, 并写入第二分数的对等值。乘以每个分数的分子和分母。划出任何共同因素, 并乘以剩余分数。

Wrap-Up: Review Questions
::总结:审查问题

roducts-and-quotients-of-rational-expressions" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5f20738e6fd9db53bb279866&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:

roducts-and-quotients-of-rational-expressions&mode=lite" test-id="5f20738e6fd9db53bb279866">

章节大纲

常规

乘和 divide 逻辑表达式

Introduction: Early Retirement ::导言:提前退休

Activity 1: Multiplying Single Term Fractions ::活动1:乘数单期分数

Activity 2: Multiplying Rational Expressions ::活动2:乘数逻辑表达式

Extension: Multiplying Rational Expressions Video ::扩展名: 视频

Activity 3: Dividing Single Term Fractions ::活动3:单期分部分

Activity 4: Dividing Rational Expressions ::活动4:分裂的理性表达式

Example ::示例示例示例示例

Extension: Dividing Rational Expressions Video ::扩展名: Dividing 理性表达式视频

Wrap-Up: Review Questions ::总结:审查问题

Introduction: Early Retirement
::导言:提前退休

Activity 1: Multiplying Single Term Fractions
::活动1:乘数单期分数

Activity 2: Multiplying Rational Expressions
::活动2:乘数逻辑表达式

Extension: Multiplying Rational Expressions Video
::扩展名: 视频

Activity 3: Dividing Single Term Fractions
::活动3:单期分部分

Activity 4: Dividing Rational Expressions
::活动4:分裂的理性表达式

Example
::示例示例示例示例

Extension: Dividing Rational Expressions Video
::扩展名: Dividing 理性表达式视频

Wrap-Up: Review Questions
::总结:审查问题