Course: 12.5 添加多面体 | The Way To Learn

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 添加多面体

Collapse Expand
添加多面体
As the students rounded the corner on Fifth Street, they spotted a peculiar looking building. It was in the shape of a pyramid. Mrs. Meery, the math teacher, asked her students the following question.
::当学生们在第五街拐角时,他们发现了一栋奇特的建筑,就像金字塔一样。数学老师Meery夫人向学生们提出了下面的问题。

“A pyramid-shaped building has rectangular floors that get increasingly smaller as you go higher up in the building. If the 87 ^th floor has a length of $\begin{align*}6x + 16\end{align*}$ and a width of 28, and each floor’s length and width decrease by 4 as you ascend, find the total area of the 87 ^th , 88 ^th , and 89 ^th floor.”
::“金字塔形建筑的长方形楼层随着你上楼而变得越来越小。如果87楼长为6x+16,宽度为28,而每层长宽则在上升时减少4,则发现87、88和89楼的总面积。 ”

In this concept, you will learn to add .
::在这一概念中,你将学会添加......。

Adding Polynomials
::添加多面体

A polynomial is an algebraic expression that shows the sum of monomials. In this concept you are going to add polynomials, but first let’s review how to add whole numbers with many digits.
::多数值是一个代数表达式,它显示了单数值的总和。在这个概念中,您要添加多数值,但首先让我们审视一下如何用多个位数来添加整数。

Add the numbers 5026 and 3210.
::加上数字5026和3210。

You might choose to add it like this.
::您可以选择这样添加它。

$\begin{aligned} 5026 \\ \underline{+ 3210} \\ 8236 \end{aligned}$
$\begin{align*}5026 \\ \underline{+3210}\\ 8236\end{align*}$

If you think about it, you might notice that the same addition could be thought of in this way.
::如果你考虑一下,你可能会注意到, 同样的添加也可以这样考虑。

$\begin{aligned} thousands hundreds tens ones \\ 5026 \to 5000 20 6 \\ \underline{+ 3210 \to + 3000 200 10} \\ 8236 \leftarrow 8000 + 200 + 30 + 6 \end{aligned}$
$\begin{align*}& \qquad \qquad \qquad \ \ \text{thousands} \quad \qquad \text{hundreds} \quad \qquad \ \text{tens} \quad \qquad \text{ones} \\ & \quad 5026 \quad \rightarrow \qquad \ \ 5000 \qquad \qquad \qquad \qquad \qquad \ \quad 20 \qquad \qquad \ \ 6\\ & \underline{+ \ 3210 \quad \rightarrow \quad \ + 3000 \qquad \qquad \ \ \ 200 \qquad \ \quad \quad \ \ \ 10 \qquad \qquad \qquad}\\ & \quad 8236 \quad \leftarrow \qquad \ \ 8000 \quad \quad \quad \ \ + 200 \quad \quad \quad \ \ \ + 30 \quad \ \quad \ \ \ \ + 6\end{align*}$
::10_8236+8000+200+30+6+6

Each of the similar places has been lined up vertically (one on top of the other) so that 3000 is beneath 5000 in the thousands place and 10 is beneath 20 in the tens place. Also, 200 is by itself because the first number had no digits in the hundreds place. Likewise, 6 is by itself because the second number had no digits in the ones place. Although this is not a practical way of writing a simple addition problem, it does demonstrate the technique you can use to add polynomials. Polynomials can be added in the same manner as we added 5026 and 3210.
::相似的位置都垂直排列( 一个在另一处 ) , 这样千个地方的 3000 低于 5000 , 10 低于 20 低于 10 。另外, 200 本身是因为第一个数字在数百个地方没有数字。同样, 6 本身是因为第二个数字在那个地方没有数字。虽然这不是写一个简单添加问题的实用方法, 但确实表明了您可以用来添加多面体的技术。多面体可以和我们加5026 和 3210 一样的方式添加。

You also need to know how to identify like terms . Like terms have exactly the same variable(s) to exactly the same power(s). When terms are alike, you can combine them by adding their coefficients.
::您也需要知道如何识别相似的术语。就像术语有完全相同的变量, 与完全相同的功率一样。当术语相同时, 您可以通过添加它们的系数将其组合在一起。

For example:
::例如:

$\begin{align*}5x^3+9x^3=14x^3\end{align*}$
::5x3+9x3=14x3

Let’s look at an example.
::让我们举个例子。

Add the polynomials $\begin{align*}(7x^2+9x-5)\end{align*}$ and $\begin{align*}(6x^2+3x+10)\end{align*}$ .
::添加多数值(7x2+9x-5)和(6x2+3x+10)

First, line up the like terms so that you can add them vertically.
::首先,排列类似术语,以便垂直添加。

$\begin{aligned} 7 x^{2} + 9 x - 5 \to 7 x^{2} + 9 x + - 5 \\ \underline{+ 6 x^{2} + 3 x + 10 \to + 6 x^{2} + 3 x + 10} \\ 13 x^{2} + 12 x + 5 \leftarrow 13 x^{2} + 12 x + 5 \end{aligned}$
$\begin{align*}& \ \ \ \ 7x^2+9x-5 \quad \rightarrow \quad 7x^2 \quad + \quad 9x \quad + \quad -5 \\ & \underline{+ \ 6x^2+3x+10 \ \ \rightarrow + \ 6x^2 \quad + \quad 3x \quad + \quad \ 10 \;} \\ & \ \ 13x^2+12x+5 \ \ \leftarrow \ \ \ 13x^2 \ \ \ + \quad 12x \ \ \ + \quad \ 5\end{align*}$
::7x2+9x-5-7x2+9x2+9xx5+6x2+3x+10 6x2+3x+10 6x2+3x+10_ 13x2+12x+5 13x2+12x+5 +5 13x2+12x+5

Each of the like terms was aligned vertically, one on top of the other. Notice that the negative sign on -5 was kept with the number 5. Be careful when you add the integers .
::相似的词词都是垂直对齐的, 一个在另一词上。请注意 - 5 上的负符号与数字 5 一致。当添加整数时要小心。

A second method for adding polynomials is horizontally—in a single line. Just as you might add $\begin{align*}6 + 19 = 25\end{align*}$ without placing them one on top of the other, polynomials can also be added horizontally .
::增加多义的第二种方法是水平化的——在单行中。您可以添加 6+19=25 而不将它们放在另一行之上,同样,也可以水平化地添加多义。

Let’s look at an example.
::让我们举个例子。

Add the polynomials $\begin{align*}(7x^2+3x-11)\end{align*}$ and $\begin{align*}(3x^2-9x+5)\end{align*}$ .
::添加多数值(7x2+3x-11)和(3x2-9x+5)

First, rewrite the polynomials without " data-term="Parentheses" role="term" tabindex="0"> parentheses . The polynomial can be rewritten without parentheses because the parentheses serve only to show the separation of the polynomials.
::首先, 重写多面体, 不带括号。多面体可以重写, 不带括号, 因为括号只能显示多面体的分隔。

$\begin{aligned} (7 x^{2} + 3 x - 11) + (3 x^{2} - 9 x + 5) = 7 x^{2} + 3 x - 11 + 3 x^{2} - 9 x + 5 \end{aligned}$
$\begin{align*}(7x^2+3x-11)+(3x^2-9x+5)=7x^2+3x-11+3x^2-9x+5\end{align*}$
:7x2+3x-11)+(3x2-9x+5)=7x2+3x-11+3x2-9x+5

Next, combine like terms.
::接下来,把术语合并起来。

$\begin{aligned} \begin{array}{rcl} (7 x^{2} + 3 x - 11) + (3 x^{2} - 9 x + 5) & = & 7 x^{2} + 3 x - 11 + 3 x^{2} - 9 x + 5 \\ = & 10 x^{2} - 6 x - 6 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (7x^2+3x-11)+(3x^2-9x+5) &=& {\color{blue}7x^2}+{\color{red}3x}-11+{\color{blue}3x^2}{\color{red}-9x}+5 \\ &=& {\color{blue}10x^2}{\color{red}-6x}-6 \end{array}\end{align*}$
:7x2+3x-11)+(3x2-9x-11)+(3x2-9x+5)=7x2+3x-11+3x2-9x+5=10x2-6)

The answer is $\begin{align*}10x^2-6x-6\end{align*}$ .
::答案是 10x2 -6x-6。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Mrs. Meery and the pyramids.
::早些时候,你被问及Meery夫人和金字塔的问题

First, write the expression for finding the area of the 87 ^th , 88 ^th , and 89 ^th floor.
::首先,写下寻找87、88和89楼面积的表达方式。

$\begin{aligned} 87^{t h} floor & : Area = 28 (6 x + 16) \\ 88^{t h} floor & : Area = (28 - 4) (6 x + 16 - 4) \\ 89^{t h} floor & : Area = (28 - 4 - 4) (6 x + 16 - 4 - 4) \end{aligned}$
$\begin{align*}87^{th} \ \text{floor} & : \text{Area} = 28(6x + 16) \\ 88^{th} \ \text{floor} & : \text{Area} = (28 - 4)(6x + 16 - 4) \\ 89^{th} \ \text{floor} & : \text{Area} = (28 - 4 - 4)(6x + 16 - 4 - 4) \end{align*}$
::87楼:Area=28(6x+16)88楼:Area=(28-4)(6x+16-4)89楼:Area=(28-4-4)(6x+16-4-4)

Next, find the areas for the three floors.
::接下来,找到三层楼的面积

$\begin{aligned} 87^{t h} floor & : Area = 28 (6 x + 16) \\ : Area = 168 x + 448 \\ 88^{t h} floor & : Area = (28 - 4) (6 x + 16 - 4) \\ : Area = 24 (6 x + 12) \\ : Area = 144 x + 288 \\ 89^{t h} floor & : Area = (28 - 4 - 4) (6 x + 16 - 4 - 4) \\ : Area = 20 (6 x + 8) \\ : Area = 120 x + 160 \end{aligned}$
$\begin{align*}87^{th} \ \text{floor} & : \text{Area} = 28(6x + 16) \\ & : \text{Area} = 168x + 448 \\ \\ 88^{th} \ \text{floor} & : \text{Area} = (28 - 4)(6x + 16 - 4) \\ & : \text{Area} = 24(6x + 12) \\ & : \text{Area} = 144x + 288 \\ \\ 89^{th} \ \text{floor} & : \text{Area} = (28 - 4 - 4)(6x + 16 - 4 - 4) \\ & : \text{Area} = 20(6x + 8) \\ & : \text{Area}= 120x + 160 \end{align*}$
::87楼:区域=28(6x+16):区域=168x+4888第层:区域=(28-4)(6x+16-4):区域=24(6x+12):区域=144x+288889第层:区域=(28-4-4)(6x+16-4-4):区域=20(6x+8):区域=120x+160

Then, find the total area for the three floors.
::然后找到三层楼的总面积

$\begin{aligned} Total area & = A_{87^{t h} floor} + A_{88^{t h} floor} + A_{89^{t h} floor} \\ = (168 x + 448) + (144 x + 288) + (120 x + 160) \\ = 432 x + 896 \end{aligned}$
$\begin{align*}\text{Total area} &= A_{87^{th} \ \text{floor}}+A_{88^{th} \ \text{floor}}+A_{89^{th} \ \text{floor}}\\ &= ({\color{blue}168x}+ 448) + ({\color{blue}144x }+ 288) + ({\color{blue}120x} + 160) \\ &={\color{blue}432x }+ 896\end{align*}$
::总面积=A87楼+A88楼+A89楼=(168x+448)+(144x+288)+(120x+160)=432x+896

The answer is $\begin{align*}432x+896\end{align*}$ .
::答案是432x+896。

The total area for the 87 ^th , 88 ^th , and 89 ^th floors $\begin{align*}432x+896\end{align*}$ is units squared.
::432x+896号第87、88和89层的总面积是平方。

Example 2
::例2

Add the polynomials $\begin{align*}(-2x^3+9x^2-3)\end{align*}$ and $\begin{align*}(8x^5+5x-14)\end{align*}$ .
::添加多面体 (-2x3+9x2-3) 和 (8x5+5x-14) 。

First, rewrite the polynomials without parentheses.
::首先,在没有括号的情况下重写多面体。

$\begin{aligned} (- 2 x^{3} + 9 x^{2} - 3) + (8 x^{5} + 5 x - 14) = - 2 x^{3} + 9 x^{2} - 3 + 8 x^{5} + 5 x - 14 \end{aligned}$
$\begin{align*}(-2x^3+9x^2-3)+(8x^5+5x-14)=-2x^3+9x^2-3+8x^5+5x-14\end{align*}$
:-2x3+9x2-3)+(8x5+5x-14)+(8x5+5x-14)\\2x3+9x2-3+9x2-3+3+8x5+5x-14)

Next, combine like terms.
::接下来,把术语合并起来。

$\begin{aligned} \begin{array}{rcl} (- 2 x^{3} + 9 x^{2} - 3) + (8 x^{5} + 5 x - 14) & = & - 2 x^{3} + 9 x^{2} - 3 + 8 x^{2} + 5 x - 14 \\ = & - 2 x^{3} + 17 x^{2} + 5 x - 17 \end{array} \end{aligned}$
$\begin{align*} \begin{array}{rcl} (-2x^3+9x^2-3)+(8x^5+5x-14) &=& {\color{green}-2x^3}+{\color{blue}9x^2}-3+{\color{blue}8x^2}+{\color{red}5x}-14\\ &=& {\color{green}-2x^3}+{\color{blue}17x^2}{\color{red}+5x}-17 \end{array}\end{align*}$
:-2x3+9x2-3)+(8x5+5x-14)+(8x5+5x-14)\\2x3+9x2-3+8x2+8x2+5x-14)_2x3+172+5x2+5x17

The answer is $\begin{align*}-2x^3+17x^2+5x-17\end{align*}$ .
::答案是-2x3+17x2+5x-17。

Example 3
::例3

Add the polynomials $\begin{align*}(4x^2+7x-2)\end{align*}$ and $\begin{align*}(3x^2+2x-1)\end{align*}$ .
::添加多数值( 4x2+7x-2) 和 (3x2+2x-1) 。

First, rewrite the polynomials without parentheses.
::首先,在没有括号的情况下重写多面体。

$\begin{aligned} (4 x^{2} + 7 x - 2) + (3 x^{2} + 2 x - 1) = 4 x^{2} + 7 x - 2 + 3 x^{2} + 2 x - 1 \end{aligned}$
$\begin{align*}(4x^2+7x-2)+(3x^2+2x-1)=4x^2+7x-2+3x^2+2x-1\end{align*}$
:4x2+7x-2)+(3x2+2x-1)=4x2+7x-2+3x2+2x-2-1)

Next, combine like terms.
::接下来,把术语合并起来。

$\begin{aligned} \begin{array}{rcl} (4 x^{2} + 7 x - 2) + (3 x^{2} + 2 x - 1) & = & 4 x^{2} + 7 x - 2 + 3 x^{2} + 2 x - 1 \\ = & 7 x^{2} + 9 x - 3 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (4x^2+7x-2)+(3x^2+2x-1) &=& {\color{blue}4x^2}+{\color{red}7x}-2+{\color{blue}3x^2}{\color{red}+2x}-1 \\ &=& {\color{blue}7x^2}{\color{red}+9x}-3 \end{array}\end{align*}$
:4x2+7x-2)+(3x2+2x-1)=4x2+7x-2+3x2+2x2+2x-1=7x2+9x-3)

The answer is $\begin{align*}7x^2+9x-3\end{align*}$ .
::答案是 7x2+9x-3。

Example 4
::例4

Add the polynomials $\begin{align*}(-4x^2+7x-2)\end{align*}$ and $\begin{align*}(-7x^2+3x-17)\end{align*}$ .
::添加多数值(- 4x2+7x-2) 和 (-7x2+3x-17) 。

First, rewrite the polynomials without parentheses.
::首先,在没有括号的情况下重写多面体。

$\begin{aligned} (- 4 x^{2} + 7 x - 2) + (- 7 x^{2} + 3 x - 17) = - 4 x^{2} + 7 x - 2 - 7 x^{2} + 3 x - 17 \end{aligned}$
$\begin{align*}(-4x^2+7x-2)+(-7x^2+3x-17)=-4x^2+7x-2-7x^2+3x-17\end{align*}$
:-4x2+7x-2)+(-7x2+3x-17)+(-7x2+3x-17)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Next, combine like terms.
::接下来,把术语合并起来。

$\begin{aligned} \begin{array}{rcl} (- 4 x^{2} + 7 x - 2) + (- 7 x^{2} + 3 x - 17) & = & - 4 x^{2} + 7 x - 2 + - 7 x^{2} + 3 x - 17 \\ = & - 11 x^{2} + 10 x - 19 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (-4x^2+7x-2)+(-7x^2+3x-17) &=& {\color{blue}-4x^2}+{\color{red}7x}-2+{\color{blue}-7x^2}{\color{red}+3x}-17 \\ &=& {\color{blue}-11x^2}{\color{red}+10x}-19 \end{array}\end{align*}$
:- 4x2+7x-2)+(- 7x2+3x- 17)+(- 7x2+3x- 3x- 17) __4x2+7x-2_ 7x2+3x- 17)_11x2+10x- 19

The answer is $\begin{align*}-11x^2+10x-19\end{align*}$ .
::答案是 - 11x2+10x-19。

Example 5
::例5

Add the polynomials $\begin{align*}(4xy+7x-2)\end{align*}$ and $\begin{align*}(-19xy-17x-9)\end{align*}$ .
::添加多边数( 4xy+7x-2) 和 (- 19xy- 17x- 9) 。

First, rewrite the polynomials without parentheses.
::首先,在没有括号的情况下重写多面体。

$\begin{aligned} (4 x y + 7 x - 2) + (- 19 x y - 17 x - 9) = 4 x y + 7 x - 2 - 19 x y - 17 x - 9 \end{aligned}$
$\begin{align*}(4xy+7x-2)+(-19xy-17x-9)=4xy+7x-2-19xy-17x-9\end{align*}$
:4xy+7x-2)+(-19xy-17x--9)=4xy+7x-2-19xy-17x-9)

Next, combine like terms.
::接下来,把术语合并起来。

$\begin{aligned} \begin{array}{rcl} (4 x y + 7 x - 2) + (- 19 x y - 17 x - 9) & = & 4 x y + 7 x - 2 + - 19 x y - 17 x - 9 \\ = & - 15 x^{2} - 10 x - 11 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (4xy+7x-2)+(-19xy-17x-9) &=& {\color{blue}4xy}+{\color{red}7x}-2+{\color{blue}-19xy}{\color{red}-17x}-9 \\ &=& {\color{blue}-15x^2}{\color{red}-10x}-11 \end{array}\end{align*}$
:4xy+7x-2)+(-19xy-17x-9)=4xy+7x-219xy-17x-915x2-10x-11)

The answer is $\begin{align*}15x^2-10x-11\end{align*}$ .
::答案是15x2-10x-11。

Review
::回顾

Add the following polynomials vertically. Be sure to align like terms.
::垂直添加以下多义。请务必对齐相似的术语。

1. $\begin{align*}(4x^2+7x-2)+(3x-17)\end{align*}$
::1. (4x2+7x-2)+(3x-17)

2. $\begin{align*}(-4x^4-x^3+8)+(-2x^3+5x+6)\end{align*}$
::2. (-4x4-x3+8)+(-2x3+5x6)

3. $\begin{align*}(10x^3-4x^2-2x+5)+(-x^2+9x-5)\end{align*}$
::3. (10x3-4x2-2x+5)+(-x2+9x-5)

4. $\begin{align*}(6x^2+5x+9)+(4x^2+3x+6)\end{align*}$
::4. (6x2+5x+9)+(4x2+3x+6)

5. $\begin{align*}(9x^2-3x+4)+(6x^2-9x+2)\end{align*}$
::5. (9x2-3x+4)+(6x2-9x+2)

6. $\begin{align*}(3y^2+4x-9)+(-5y^2-6x+10)\end{align*}$
::6. (3y2+4x-9)+(-5y2-6x+10)

7. $\begin{align*}(14x^2+6x-2)+(9x-1)\end{align*}$
::7. (14x2+6x-2)+(9x-1)

8. $\begin{align*}(-2x^2+7x-2)+(-3x^2-17)\end{align*}$
::8. (-2x2+7x-2)+(-3x2-17)

9. $\begin{align*}(9x^2+7x-2y)+(3x^2-x+9y)\end{align*}$
::9. (9x2+7x-2y)+(3x2-x+9y)

10. $\begin{align*}(4xy+7x-21)+(-12xy+4x-8)\end{align*}$
::10. (4xy+7x-21)+(-12xy+4x-8)

11. $\begin{align*}(11x^2+9x-2y)+(3x^2-8x-5y-2)\end{align*}$
::11. (11x2+9x-2y)+(3x2-8x-5y-2)

Add the following polynomials horizontally.
::水平添加以下多义。

12. $\begin{align*}(-3x-8)+(15x+5)\end{align*}$
::12. (-3x-8)+(15x+5)

13. $\begin{align*}(x^4+7x^3-2x+7)+(-8x^3+9x^2-4)\end{align*}$
::13. (x4+7x3-2x7)+(-8x3+9x2-4)

14. $\begin{align*}(4x^2y-3x^2y^2+7xy)+(9x^2y^2-5xy+3x^2)\end{align*}$
::14. (4x2y-3x2y2+7xy)+(9x2y2-5xy+3x2)

15. $\begin{align*}(5xy-3x+19)+(4xy-9x-22)\end{align*}$
::15. (5xy-3x+19)+(4xy-9x-22)

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。

Resources
::资源

Section outline

General

添加多面体

Adding Polynomials ::添加多面体

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Adding Polynomials
::添加多面体

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源