1.1 多面性属性-interactive

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  多面性属性

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    Learning Objectives
  ::学习目标

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  • Understand the characteristics and structure of .
    ::了解......的特点和结构。
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  • Convert a polynomial to standard form .
    ::将多边形转换为标准窗体。
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  • Identify the degree of a polynomial.
    ::识别一个多面性的程度 。

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  Introduction:  Does Technology Involve  Algebra?
  ::导言:技术涉及代数吗?

  

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  Algebra is a powerful tool that uses numbers, variables, expressions, equations, and functions to describe the world around you. By describing the world in mathematical terms , humans have developed a better understanding of how it works, leading to advancements in robotics, computing, self-driving cars, video games, 3-D printers, smartphones, and all of the everyday technology that tends to be taken for granted. All of these examples use mathematics to recreate real-world characteristics. Algebra will offer you the ability to interact with the world in new and fantastic ways.
  ::代数是一个强大的工具,它使用数字、变量、表达式、方程式和函数来描述你周围的世界。通过用数学术语描述世界,人类已经更好地了解了世界如何运作,从而导致机器人、计算、自驾汽车、电子游戏、三维打印机、智能手机和所有日常技术的进步,这些都倾向于理所当然地使用数学来重建现实世界的特征。代数将为您提供与世界进行新颖和奇妙互动的能力。

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  Before you  can begin modeling the world around you, you will first need to understand basic polynomial vocabulary and operations . Cell phones, video games, computers, and more all use polynomials to transmit information. When you send someone a text message that says “Hi,” you just sent them a polynomial. The characters in the message are converted to the binary number system, which has the properties of a polynomial. For example, if a message contains the lower case letter a, this would be written in binary as 01100001. When converted to polynomial form, it looks like this :
  ::在您开始模拟周围的世界之前, 您首先需要理解基本的多边词汇和操作 。 手机、 视频游戏、 计算机, 以及更多的人使用多边词汇来传递信息 。 当您发送了一个写着“ Hi” 的文本信息时, 您只是给他们发了一个多义 。 信件中的字符被转换为二进制数字系统, 它具有多式的属性 。 例如, 如果信件包含小写字母a , 它会以 01100001 的形式以二进制写入 。 当转换成多边词汇形式时, 它看起来像 :

  

   

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  The digit farthest to the right is assigned the variable  $\begin{array}{r}{x}^0,\end{array}$  and each following digit from right to left is assigned a variable with a power that is one greater than the previous power. The final polynomial is written without the variables that have coefficients of zero. Throughout this chapter, you will take an in-depth look at how cell phones and computers use polynomial operations to send information from one device to another.
  ::向右的位数最远被指定变量x0, 从右到左的位数各被指定一个变量,该变量的功率大于先前的功率。最后的多数值是没有零系数的变量写入的。在整个本章中,您将深入查看手机和计算机如何使用多数值操作将信息从一个设备发送到另一个设备。

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  Use the interactive below to convert letters from binary to polynomial form.
  ::使用下面的交互效果将字母从二进制转换为多式格式。

  

  INTERACTIVE

  Using Algebra

  

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  • Choose a letter from the dropdown. Press the Next button to see its representation in binary.
    ::从下拉键中选择信件。 按下下一个按钮以查看其二进制表达式 。
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  • Use the dropdowns to convert the binary number into a polynomial.
    ::使用下拉键将二进制数字转换为多式数字 。
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  • Check your answer with the Check button.
    ::用检查按钮检查您的回答 。
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  • Once correct, try another with the New Letter button. 
    ::一旦正确, 请用新字母按钮再试一次 。

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  Activity 1: What is a Polynomial?
  ::活动1:什么是一夫多妻制?

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  A polynomial is an expression that consists of terms connected by addition or subtraction . A term consists of a real number  coefficient , multiplied by a variable , raised to a power that is a natural number . The coefficient and variables that make up a term are known as its factors .
  ::多式表达式由增加或减法相联的术语组成。一个术语由实际数字系数乘以变量组成,形成自然数字的力量。构成一个术语的系数和变量被称为其因素。

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  Use the interactive below to  practice identifying  expressions that are polynomials.
  ::使用以下互动方式来练习识别多面性表达式。

  olynomials-in-standard-form" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5eb6c9bd400a1bb551940e4e&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:olynomials-in-standard-form&mode=lite" test-id="5eb6c9bd400a1bb551940e4e">

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  Activity 2: Properties of Polynomials
  ::活动2:多面性属性

  

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  A polynomial is  commonly defined by degree and number of terms. The leading term of a polynomial can typically be identified as the term with the greatest exponent value. The exponent in this term is referred to as the degree, and the coefficient is referred to as the leading coefficient .
  ::多元性通常按程度和数量界定。多元性的主要术语通常可以被确定为具有最大指数值的术语。这一术语的引号被称为程度,系数则被称为主要系数。

  

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  A polynomial is considered to be in standard form if the terms are written by decreasing order of degree.
  ::如果术语按降级顺序排列,则多语种被视为标准形式。

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  Use the interactive below to identify characteristics and put the following .
  ::使用下面的交互式文字来辨别特征,并写下以下文字:

  

  INTERACTIVE

  Standard Form

  

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  • Practice writing a polynomial in standard form.
    ::练习以标准格式写多面体。
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  • After choosing a game mode, use the red points to re-order the terms in standard form.
    ::选择游戏模式后, 使用红点来按标准格式重新排序条件 。

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  Use the interactive below to practice identifying properties of polynomials.
  ::使用下面的交互功能来练习多面形特性的识别。

  

  INTERACTIVE

  Monomials, Binomials, Trinomials: Classification

  

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  What is the difference between a monomial, binomial and trinomial? Help classify these polynomials. 
  ::单血球、二血球和三血球之间有什么区别? 帮助分类这些多血球。

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  • Select a label, then select the cards that belong to that category.
    ::选择标签,然后选择属于该类别的卡片。

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  olynomials-in-standard-form" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5eb6de07f0bde57dd3a0518b&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:olynomials-in-standard-form&mode=lite" test-id="5eb6de07f0bde57dd3a0518b">

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  Activity 3: Writing Polynomials
  ::活动3:书写多面体

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  V ariables  represent  unknown quantities in expressions and  mathematical  relationships. Many real-world relationships can be modeled using variables in a polynomial. This practice is commonly seen in geometry and geometric applications. Engineers, architects, carpenters, fashion designers, packaging designers, and many more professions use software that helps apply these relationships to model complicated design patterns. The use of polynomial expressions allows you to change values  (represented by variables) without having to worry about  completely rewriting the expression each time.
  ::变量代表表达式和数学关系中未知的数量。 许多真实世界关系可以使用多数值变量进行模型化。 这种做法通常见于几何和几何应用中。 工程师、 建筑师、 木匠、 时装设计师、 包装设计师和更多的专业使用软件来帮助应用这些关系来模拟复杂的设计模式。 使用多数值表达式可以使您改变值( 以变量表示 ) , 而不必担心每次完全重写表达式。

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  Example : Pool and Walkway
  ::示例:游泳池和人行道

  

  

   

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  The images above  show  an architectural plan to build a paved walkway around a square pool. The amount of concrete needed will depend on the area of the walkway. Write a polynomial to represent the area of the paved walkway.
  ::上面的图象显示一个建筑计划,在广场池周围建造一个铺面的行走道。 需要的混凝土数量将取决于行走道的面积。 写一个多面体来代表铺面行走道的面积 。

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  To write the polynomial, break the walkway up into smaller sections:
  ::要写多义, 将行进道分割成小段 :

  

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  To find the area of the paved section, you simply need to add up the areas of smaller sections that make up the walkway. The area of both a square and rectangle can be found by multiplying the length by the width. This product will give you the following polynomial:
  ::要找到铺面段的面积, 您只需将行道上小段的面积相加即可。 方形和矩形的面积都可以通过将长度乘以宽度来找到。 此产品将为您提供以下多面性 :

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  Answer:   $\begin{array}{r}{x}^2+10x+x^2+10x+x^2+10x+x^2+10x\end{array}$
  ::答复:x2+10x+x2+10x+x2+10x+10x+10x+x2+x2+10x

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  The term  $\begin{array}{r}{x}^2\end{array}$  represents the square corners of the paved walkway, and the term 10x represents the long, narrow, rectangular sections of walkway, each the same length as the side of the pool . Is this the only way you could have written the polynomial? Can you think of any other ways to write it?
  ::x2 表示铺面行道的角, 10x 表示长、 窄、 长、 长、 长、 长、 长、 长、 长、 长、 长、 长, 长度与 池边相同 。 这是您唯一可以写出多面行道的方法吗 ? 您能想到其他写法吗 ?

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  Example : Event Attendance
  ::实例:出席活动

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  Write a polynomial to represent the average attendance at three different events.
  ::撰写一个多义词, 以代表三次不同活动的平均出席人数 。

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  The average attendance at three different events can be modeled as follows:
  ::三次不同活动的平均出席人数可模拟如下:

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  $$\begin{array}{r}\frac{a₁+a₂+a₃}{3}\end{array}$$

  
  ::123a1+a2+a33

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  Although this may not immediately look like a polynomial, recall that you can rewrite a fractional expression as each term in the numerator over the denominator.
  ::虽然这也许不会立即看起来像一个多符号, 但要记住, 您可以重写分数表达式, 作为分母上的分子中的每个词 。

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  $$\begin{array}{rl}\frac{a₁+a₂+a₃}{3}& =\frac{a₁}{3}+\frac{a₂}{3}+\frac{a₃}{3}\end{array}$$

  
  ::123123a1+a2+a33=a13+a23+a33

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  Answer: $\begin{array}{r}\frac{1}{3}a₁+\frac{1}{3}a₂+\frac{1}{3}a₃\end{array}$  
  ::答复:12313a1+13a2+13a3

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  The variables are the attendance, and the coefficients are each  $\begin{array}{r}\frac{1}{3}\end{array}$  because each attendance represents a third of the average.
  ::变量是出勤率,系数是每13个系数,因为每次出勤率为平均数的三分之一。

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  Guided Practice : The questions  below include feedback to help you understand why your answers are correct or incorrect.  
  ::指导实践:下列问题包括反馈,帮助你了解答案为何正确或不正确。

  olynomials-in-standard-form" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5eb6e639dc1be270190daed3&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:olynomials-in-standard-form&mode=lite" test-id="5eb6e639dc1be270190daed3">

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  Wrap-Up: Review Questions
  ::总结:审查问题

  olynomials-in-standard-form" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5eb6e9dbfc0522caccf4ca19&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:olynomials-in-standard-form&mode=lite" test-id="5eb6e9dbfc0522caccf4ca19">

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  Summary
  ::摘要

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  • A polynomial is an expression that consists of terms connected by addition or subtraction.
    ::多式表达式是由增加或减法连接的术语组成的表达式。
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  • Each term has a variable raised to a power that is a natural number.
    ::每个术语都有一个变量 提升到自然数的功率
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  • The standard form of a polynomial  has terms written by decreasing order of degree.
    ::标准形式的多民族语言有按程度递减的文字。