Section outline

  • General

    Collapse all Expand all

  • 添加和减减多面体

    You work two jobs —as waitstaff a couple of nights during the week and as a cashier on the weekends. The waitstaff job pays $5.50 per hour and you make about $50 in tips per night, and the cashier jobs pays $9.50 per hour, you can write a polynomial model to describe your pay each week by adding polynomial models for each job together. In this section, we discuss how to add and subtract .
    ::你从事两个工作——一周内做两个晚上的等待工作人员,周末做一个出纳员。等待工作人员的工作每小时支付5.50美元,每小时支付50美元的小费,而出纳员的工作每小时支付9.50美元。您可以写一个综合模型来描述每周的工资,为每份工作加一个多面性模型。在本节中,我们讨论如何增减。

    lesson content

     

    Adding Polynomials
    ::添加多面体

    Since addition is commutative (the order we add terms in does not effect the final result), we can rearrange a sum of two polynomials so that the like terms are next to each other. Recall, like terms are terms with the same variables and each variable is raised to the same power. After we rearrange the terms, then we combine like terms.
    ::由于添加为通量( 我们添加的顺序不影响最终结果) , 我们可以重新排列两个多义词的总和, 这样相似的词可以相邻。 回顾, 类似术语是同变量的术语, 并且每个变量会被提升到相同的力量 。 在我们重新排列术语后, 我们将类似术语合并 。

    As we will see below, you can organize your work horizontally or vertically .
    ::如下文所示,您可以横向或纵向组织您的工作。

    Example 1
    ::例1

    Simplify ( 4 x 3 − 2 x 2 + 4 x + 15 ) + ( x 4 − 8 x 3 − 9 x − 6 )
    ::简化 (4x3 - 2x2+4x+15)+( x4 - 8x3 - 9x-6)

    Solution: To add or subtract two polynomials, combine like terms. In the horizontal set-up, we regroup the terms in the polynomial to bring together  the like terms. In the vertical set-up, we stack the like terms in the same columns.
    ::解决方案 : 要添加或减去两个多面体, 将相似的术语组合在一起。 在水平设置中, 我们将多面体中的术语重新组合, 将相似的术语组合在一起。 在垂直设置中, 我们将相似的术语堆叠在同一列中 。

    ( 4 x 3 − 2 x 2 + 4 x + 15 ) + ( x 4 − 8 x 3 − 9 x − 6 ) = x 4 + ( 4 x 3 − 8 x 3 ) − 2 x 2 + ( 4 x − 9 x ) + ( 15 − 6 ) = x 4 − 4 x 3 − 2 x 2 − 5 x + 9 OR 4 x 3 − 2 x 2 + 4 x + 15 x 4 − 8 x 3       − 9 x − 6 _ x 4 − 4 x 3 − 2 x 2 − 5 x + 9

    :sad4x3 - 2x2+4x+4x+15)+(x4 - 8x3 - 8x3 - 9x-6)=x4+(4x3 - 8x3 - 8x3)-2x2+(4x3 - 9x3)+(4x - 9x)+(4x4 - 9x)+(15 - 6)+(4x4 - 4x3 - 4x3 - 2x)+(15 - 6)+(4x - 6)+(4x4 - 4x3 - 2x)+(x4 - 4x3 - 4x3 - 2x2 - 5x)+(x4 - 4x3 - 4x3 - 4x3 - 2x2x2 - 5x) - 5x+2x2 - 9OR4x3 - 2x2 - +2x2x4x4x2 - +4x4x4x4+4x15x4x4x4 - 8x3 - 8x3 - 9x6-9x6x3 - 9x4x3 -4x3 -4x2x2x2x2x2x2x2x2x2x2 -5x-5x-5x

    by CK-12 demonstrates how to add polynomials. 
    ::使用 CK-12 演示如何添加多语种。

     

     

    by CK-12 takes a vertical approach to adding polynomials. 
    ::使用 CK-12 采用垂直法添加多面体 。

     

    Subtracting Polynomials
    ::减 减 多元数

    When we subtract integers , we keep the sign of the first number, change the operation to addition, and change the sign of the second number. For example,
    ::当我们减去整数时,我们保留第一个数字的符号,将操作改为添加,并更改第二个数字的符号。例如,

    − 13 − 17 = − 13 + − 17 = − 30

    Since variable expressions ultimately represent numbers, we are going to use the same technique here to change subtraction problems to addition problems. One important thing to note is, unlike the second number above that is only one term , polynomials often have multiple terms, so it is necessary to change the sign of  all of the terms in the second polynomial. You can think of this as distributing a -1 to each term in the second polynomial. 
    ::由于变量表达式最终代表数字, 我们将使用相同的技术来将减法问题转换为添加问题。 需要注意的一点是, 与上面第二个数字( 仅一个词) 不同, 多语种通常有多个术语, 因此有必要更改第二个多语种中所有术语的符号。 您可以将此视为在第二个多语种中将 - 1 分配到第二个多语种中的每个术语 。

    Like addition, we can set these up horizontally or vertically. 
    ::此外,我们可以横向或纵向设置这些装置。

    Example 2
    ::例2

    Simplify ( 2 x 3 + x 2 − 6 x − 7 ) − ( 5 x 3 − 3 x 2 + 10 x − 12 )
    ::简化 (2x3+x2- 6x- 7)- (5x3- 3x2+10x- 12)

    Solution: When subtracting, distribute the negative sign to every term in the second polynomial, then combine like terms.
    ::解决方案: 减去时, 将负符号分布到第二个多义词中的每个词, 然后将类似术语合并 。

    ( 2 x 3 + x 2 − 6 x − 7 ) − ( 5 x 3 − 3 x 2 + 10 x − 12 )
    :sad2x3+x2-6x-7)-(5x3-3-3x2+10x-12)

    Vertically, we have
    ::垂直,我们有

    ( 2 x 3 + x 2 − 6 x − 7 ) − ( 5 x 3 − 3 x 2 + 10 x − 12 ) = 2 x 3 + x 2 − 6 x − 7 − 5 x 3 + 3 x 2 − 10 x + 12 = ( 2 x 3 − 5 x 3 ) + ( x 2 + 3 x 2 ) + ( − 6 x − 10 x ) + ( − 7 + 12 ) = − 3 x 3 + 4 x 2 − 16 x + 5
       
    :sad2x3+x2 - 6x-2 - 6x-7)-(5x3 - 3x2+10x-2- 10x-12)=2x3+6x2 - 7-7 - 5x3+3x2 - 10x2+12=(2x3 - 5x3)+(x2+3x2+3x2+3x2+10x2)+(-6x- 10x)+(-7+12)_3x3+4x2 - 16x5)

    by CK-12 demonstrates how to subtract polynomials. 
    ::通过 CK-12 演示如何减去多数值。

     

     

    by CK-12 takes a vertical approach to subtracting polynomials.
    ::在 CK-12 中,对减减多数值采用垂直法。

     

     

       WARNING
    ::警告

    Distribute -1 to each one of the terms in the second polynomial, not just the first one. The following is incorrect. 
    ::将 - 1 分配到第二个多义词中的每一个词, 而不仅仅是第一个词。 以下不正确 。

    ( 3 x 2 + 2 x − 1 ) − ( x 2 + 2 x + 4 ) ≠ 3 x 2 + 2 x − 1 + − x 2 + 2 x + 4

    :sad3x2+2x-1)-(x2+2x+4) 3x2+2x-1*x2+2x-1*x2+2x+4)

    Example 3
    ::例3

    You work two jobs—as waitstaff a couple of nights during the week and as a cashier on the weekends. The waitstaff job pays $5.50 per hour and you make about $50 in tips per night and the cashier jobs pays $9.50 per hour. If you work 20 hours a week between the two jobs, write a polynomial model to describe your pay each week. 
    ::你从事两个工作——一周内担任两个晚上的值班员,周末担任出纳员,每小时支付5.50美元,每晚支付50美元小费,每晚支付9.50美元,每小时支付9.50美元,如果在两份工作之间每周工作20小时,请写一个综合模型,说明每周的工资。

    Solution:   Let h be the number of hours you wait tables each week. Then, the number of hours you work as a cashier is 20- h. The pay for each of your jobs can be modeled as below.
    ::解答: 请将您每周的等待表的小时数列为小时数。 然后, 您作为收银员的工作小时数为 20 小时。 您的每一项工作的工资可以按以下模式来计算。

    %3D5.5h%2B100%5C%5C%0A%0A%26Cheart%3D9.5(20-h)%3D190-9.5h">

    W ( h ) = 5.5 h + 100 C ( h ) = 9.5 ( 20 − h ) = 190 − 9.5 h

    ::Wheart=5.5h+100Cheart=9.5(20-h)=190-9.5h

    In the first function , we add 100 for the tips that you make on two nights per week. In the second function, we distribute the 9.5 to each one of the terms.  
    ::在第一个函数中,我们加100美元作为你每周两晚的提示。在第二个函数中,我们将9.5分用于每个术语。

    Your total paycheck is   %2BCheart%3D5.5h%2B100%2B190-9.5h%3D(5.5h-9.5h)%2B(100%2B190)%3D-4h%2B290."> W ( h ) + C ( h ) = 5.5 h + 100 + 190 − 9.5 h = ( 5.5 h − 9.5 h ) + ( 100 + 190 ) = − 4 h + 290.  
    ::您的薪酬总额是 Wheart+Cheart=5.5h+100+190-9.5h=(5.5h-9.5h)+(100+190)+4h+290。

    by CK-12 demonstrates how to add and subtract polynomials. Several examples are provided. 
    ::CK-12 演示了如何增减多数值,提供了几个例子。

     

    Summary
    ::摘要

    • To add polynomials, identify the like terms and add their coefficients.
      ::添加多语种, 标明类似术语, 并添加系数 。
    • To subtract polynomials, distribute -1 to each term in the second polynomial, then follow the steps for addition.
      ::要减去多面体,则在第二个多面体中将 - 1 分配到每个术语,然后遵循添加步骤。

    Review
    ::回顾

    Add or subtract the following polynomials.
    ::添加或减去下列多数值。

    1.  ( x 3 + 8 x 2 − 15 x + 11 ) + ( 3 x 3 − 5 x 2 − 4 x + 9 )
    ::1. (x3+8x2-15x+11)+(3x3-5x2-4x+9)

    2.  ( − 2 x 4 + x 3 + 12 x 2 + 6 x − 18 ) − ( 4 x 4 − 7 x 3 + 14 x 2 + 18 x − 25 )
    ::2. (-2x4+x3+12x2+6x-18)-(4x4-7x3+14x2+18x-25)

    3.  ( 10 x 3 − x 2 + 6 x + 3 ) + ( x 4 − 3 x 3 + 8 x 2 − 9 x + 16 )
    ::3. (10x3-x2+6x+3)+(x4-3x3+8x2-9x+16)

    4.  ( 7 x 3 − 2 x 2 + 4 x − 5 ) − ( 6 x 4 + 10 x 3 + x 2 + 4 x − 1 )
    ::4. (7x3-2x2+4x-5)-(6x4+10x3+x2+4x-1)

    5.  ( 15 x 2 + x − 27 ) + ( 3 x 3 − 12 x + 16 )
    ::5. (15x2+x-27)+(3x3-12x+16)

    6.  ( 2 x 5 − 3 x 4 + 21 x 2 + 11 x − 32 ) − ( x 4 − 3 x 3 − 9 x 2 + 14 x − 15 )
    ::6. (2x5-3x4+21x2+11x-32)-(x4-3x3-9x2+14x-15)

    7.  ( 0.56 x 5 + 1.72 x 3 − 4.73 x 2 ) + ( 1.29 x 4 − 3.59 x 3 + 6.32 x 2 )
    ::7. (0.56x5+1.72x3-4.73x2+1.29x4-3.59x3+6.32x2)

    8.  ( 1 4 x 3 − 2 3 x 2 + 4 5 x ) − ( 5 8 x 3 + 7 9 x 2 + 4 15 x )
    ::8. (14x3-23x2+45x)-(58x3+79x2+415x)

    9.  ( 8 x 3 − 13 x 2 + 24 ) − ( x 3 + 4 x 2 − 2 x + 17 ) + ( 5 x 2 + 18 x − 19 )
    ::9. (8x3-13x2+24)-(x3+4x2-2x+17)+(5x2+18x-19)

    10.  ( − 7 x 5 + 8 x 4 + 11 ) − ( − 5 x 5 + 3 x 3 − 12 ) − ( − 6 x 4 + 7 x 3 − 7 )
    ::10. (-7x5+8x4+11)-(-5x5+3x3-12)-(-6x4+73-7)

    Explore More
    ::探索更多

    1. You run two businesses, one with a revenue of  − x 2 + 5 x + 20 , 000  and another with a revenue of  − 3 x 2 + 7 x + 15 , 000 . What is your total revenue? What is the difference between the revenues for your two businesses? 
    ::1. 你经营两家企业,一家收入为-x2+5x+20,000,另一家收入为-3x2+7x+15000。你的总收入是多少?你两家企业的收入之间有什么差别?

    2. Find the perimeter of a rectangular frame with sides of  2 x − 5  and  3 x + 4 .
    ::2. 寻找长方形框架的边缘,其侧面为2x-5和3x+4。

    3. Suppose that two cars are racing . The distance traveled by one car after t seconds is 10 t 2 + 50 t  meters, while the distance traveled by the other car after t seconds is 15 t 2 + 40 t   meters. How far would the two cars be apart after t seconds?  
    ::3. 假设有两辆汽车在赛车,一辆车在2秒后行驶的距离是10t2+50米,另一辆车在2秒后行驶的距离是15t2+40米,两辆汽车在2秒后相隔多远?

    Answers for Review and Explore More Problems
    ::回顾和探讨更多问题的答复

    Please see the Appendix.  
    ::请参看附录。

    PLIX
    ::PLIX

    Try these interactives that reinforce the concepts explored in this section:
    ::尝试这些强化本节所探讨概念的交互作用 :