Section outline

  • General

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  • 正在添加和减减多边和减法

    Learning Objectives
    ::学习目标

    • Understand that form a system analogous to the integers , namely, they are closed under the operations of addition and subtraction .
      ::理解它会形成一个与整数相似的系统,即,在增减操作下关闭。
    • Add and subtract polynomials within a real-world context.
      ::在现实世界背景下添加和减去多数值。

    Introduction: Modeling in the Business World
    ::导言:商业界的建模

    lesson content

    Any integer can be represented as a polynomial . Since polynomials can be used to represent numbers, this means that they can be added, subtracted, multiplied, and divided. Some of these applications might seem obvious, like adding and subtracting lengths, whereas others may not be as apparent , such as combining rates. In this section,  you will explore real-world situations that require adding and subtracting polynomials.
    ::任意整数都可以以多数值表示。 由于多数值可以用来表示数字, 这意味着它们可以被添加、减、增和除。 其中某些应用程序可能看起来显而易见, 比如增和减长度, 而其他应用程序可能不那么明显, 比如合并率。 在本节中, 您将探索需要增和减多数值的现实世界状况 。

    One of the keys to running a successful business is predicting sales in advance. However, how can you predict revenue , cost, and profit with so many unknowns? These situations are where variables and polynomials come in handy. T hese quantities can be modeled in advance of knowing specific values.  Once  you have the values, it will be much easier to plug them in and understand how they will affect the overall quantity. Use the interactive below to practice writing simple polynomial expressions from business contexts before combining them.
    ::成功经营企业的关键之一是提前预测销售额。 但是, 您如何预测收入、 成本和利润? 这些是变量和多数值的有用之处。 这些数量可以在了解特定值之前进行模型化。 一旦您掌握了这些值, 就可以更容易地插入这些值, 并了解它们会如何影响总数量 。 使用下面的交互功能, 在合并这些值之前先从商业环境中进行简单的多数值表达 。

      

    Activity 1: Adding Polynomials
    ::活动1:增加多面体

    Before adding polynomials, let’s review how to add integers considering that polynomials represent integers, and integers can be written in a polynomial form.
    ::在添加多数值之前,让我们先审查如何添加整数,因为多数值代表整数,而整数可以用多数值的形式写出。

    lesson content

    A pply the same approach to adding polynomials: lining them up vertically by like terms .
    ::使用相同的方法来添加多种语言: 以类似术语垂直排列它们 。

    Example
    ::示例示例示例示例

    An online business manufactures and sells two brands of portable speakers: the Eqo and the Impulse. The projected yearly revenue for each couch can be modeled using a polynomial where x is the number of years after 2020. The projected revenue, in thousands, of the Eqo can be modeled using the polynomial 0.001 x 3 − 0.6 x 2 + 9.2 x − 32.8 . The projected revenue, in thousands, of the Impulse can be modeled using the polynomial 0.38 x 2 + 37 x + 200.  Write a polynomial to represent the total revenue from both speakers.
    ::在线商业制造和销售两品牌便携式喇叭:Eqo和Umpulse。每张沙发的预计年收入可以使用一个多面体模拟,其中x是2020年后的年份。Eqo的预计收入(以千计)可以使用多面体0.001x3-0.6x2+9.2x-32.8进行模拟。Inpulse的预计收入(以千计)可以使用多面体0.38x2+37x200进行模拟。写一个多面体模拟,以代表两位发言者的总收入。

    To find the total revenue, add the revenues of both speakers. For this example, let’s line the polynomials up vertically to reinforce the idea of like terms.
    ::为了找到总收入,请加上两位演讲者的收入。 比如,让我们垂直地将多语言学校排成一行,以强化类似术语的概念。

    0.001 x 3 − 0.6 x 2 + 9.2 x − 32.8 + 0.38 x 2 + 37 x + 200 0.001 x 3 − 0.22 x 2 + 46.2 x + 167.2

    ::001x3-0.6x2+9.2x-32.8+0.38x2+37x2000.001x3-0.22x2+46.2x+167.2

    Answer : The combined revenue of the two portable speakers can be modeled using the polynomial  0.001 x 3 − 0.22 x 2 + 46.2 x + 167.2 ,  where x  is the number of years after 2020.
    ::答复:两个便携式喇叭的综合收入可使用多元0.001x3-0.22x2+46.2x+167.2来模拟,其中x为2020年之后的年数。

    When adding polynomials, combine like terms. It may help visually to line the polynomials up vertically, but this step is not necessary. If you choose to line the polynomials up vertically, be sure to simplify the polynomials if possible before adding.
    ::添加多义时, 将类似术语合并。 直观地将多义线垂直直线可能有帮助, 但此步骤没有必要 。 如果您选择将多义线垂直直线, 请务必在添加前尽可能简化多义 。

      

    Activity 2: Subtracting  Polynomials
    ::活动2:减 减 数 数 数 数 数

    C onsider how  to subtract integers before subtracting polynomials.
    ::在减去多数值之前考虑如何减去整数 。

    lesson content

    The process for subtracting polynomials is similar to the process for adding polynomials. The only difference is that the terms of the expression that is being subtracted.
    ::减减多数值的过程与加增多数值的过程相似。 唯一的区别在于要减减的表达式条件 。

    Example
    ::示例示例示例示例

    Subtract  ( 7 x 4 − 10 x 2 − 32 x ) − ( x 3 − 26 x 2 − 3 x + 17 )
    ::减号(7x4 - 10x2 - 32x)-(x3 - 26x2 - 3x+17)

    7 x 4 − 10 x 2 − 32 x − ( x 3 − 26 x 2 − 3 x + 17 ) 7 x 4 − x 3 + 16 x 2 − 29 x − 17

    ::7x4 - 10x2 - 32x- (x3 - 26x2 - 3x+177x4 - x3 - 16x2 - 29x-17)

    Answer:   7 x 4 − x 3 + 16 x 2 − 29 x − 17
    ::答复:7x4-x3+16x2-29x-17

    Notice that when subtracting a polynomial, the whole polynomial is being subtracted. P olynomials  can be subtracted by lining them up vertically, but the subtraction symbol must be  distributed before combining like terms.
    ::请注意, 当减去多数值值时, 整个多数值值将被减去。 多数值值可以通过垂直排列来减去, 但减值符号在合并类似条件之前必须分布 。

    INTERACTIVE
    Subtracting Polynomials
    minimize icon

    A step-by-step walk-through of how to subtract polynomials.
    ::一步一步地走过去 如何去掉多面体

    • Click the start button to begin
      ::单击启动按钮开始
    • Shift through the steps using the arrows
      ::使用箭头在阶梯上移动
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    Activity 3:  Modeling in the Business World Continued
    ::活动3:在商业界继续建模

    A company will be making and selling a new video game console called the X-Station. A market analysis was performed to find the ideal price as well as potential revenues, costs, and profits. The following are the results of the analysis: 
    ::一家公司将制作和销售一个名为X级的新视频游戏控制台。 进行了市场分析,以找到理想价格以及潜在收入、成本和利润。 以下是分析结果:

    The analysis reported that the polynomial  47 n − n 2  represents the price at which n consoles, in thousands, will be sold. For example, if the company wants to sell 10,000 consoles, they should set the price to $370.
    ::分析报告指出,多面性47n-n2是N控制台(以千计)出售的价格。 比如,如果公司想出售10,000个控制台,它们应该将价格定在370美元。

    47 ( 10 ) − ( 10 ) 2 = 370

    The monthly cost, in thousands, of making n consoles, in thousands, can be modeled using  %20%3D%20239x%2B287."> C ( n ) = 239 x + 287.
    ::用CNo=239x+287模型制作N控制台每月成本(千),以千计。

     

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

     

    Extension: Interactive Practice
    ::扩展:交互式做法

    INTERACTIVE
    Adding and Subtracting Polynomials
    minimize icon
    • Click the cards that represent terms of the polynomial to change their coefficients
      ::单击代表多数值条件的卡片以改变其系数
    • Click the card with the operator to switch between adding and subtracting the red and blue polynomials
      ::单击用操作员切换的卡,在添加和减去红色和蓝色多元数值之间切换
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    Summary
    ::摘要

    • To add or subtract polynomials, line them up vertically by like terms.
      ::要添加或减去多数值,请按类似条件垂直排列它们。
    • When subtracting polynomials, distribute the subtraction symbol before combining like terms.
      ::减去多数值时,在合并类似条件之前,先分配减号。
    • If a polynomial is missing a term, add in the missing term with a coefficient of zero ( 0 x n ) .  
      ::如果缺少一个词汇,则在缺少的词汇中加上0xn的系数(0xn)。