Section: 聚合物加和减减 | 9.2 多元性增加和减法

Section outline

Addition and Subtraction of Polynomials
::聚合物加和减减

To add two or more , write their sum and then simplify by combining like terms .
::加上两个或两个以上,写下其数额,然后将类似术语合并简化。

Adding Polynomials
::添加多面体

Add and simplify the resulting polynomials.
::添加并简化由此产生的多面体。

a) Add $\begin{aligned} 3 x^{2} - 4 x + 7 \end{aligned}$ and $\begin{aligned} 2 x^{3} - 4 x^{2} - 6 x + 5 \end{aligned}$
:a) 添加3x2-4x+7和2x3-4x2-6x+5

$\begin{aligned} (3 x^{2} - 4 x + 7) + (2 x^{3} - 4 x^{2} - 6 x + 5) \\ Group like terms: & = 2 x^{3} + (3 x^{2} - 4 x^{2}) + (- 4 x - 6 x) + (7 + 5) \\ Simplify: & = 2 x^{3} - x^{2} - 10 x + 12 \end{aligned}$

:3x2 - 4x+7)+(2x3 - 4x2 - 4x2 - 6x+5) +(2x3 - 4x2 - 4x2)+(4x - 6x6x)+(7+5)+(2x3 - 4x2 - 6x+5) +(2x3 - 4x2 - 6x+5) 类似词组:=2x3+(3x2 - 4x2 - 4x2)+(4x - 6x)+(7+5)+(7+5)+(简化:=2x3 - x2 - 10x+12)

b) Add $\begin{aligned} x^{2} - 2 x y + y^{2} \end{aligned}$ and $\begin{aligned} 2 y^{2} - 3 x^{2} \end{aligned}$ and $\begin{aligned} 10 x y + y^{3} \end{aligned}$
:b) Add x2-2xy+y2和2y2-3x2和10xy+y3

$\begin{aligned} (x^{2} - 2 x y + y^{2}) + (2 y^{2} - 3 x^{2}) + (10 x y + y^{3}) \\ Group like terms: & = (x^{2} - 3 x^{2}) + (y^{2} + 2 y^{2}) + (- 2 x y + 10 x y) + y^{3} \\ Simplify: & = - 2 x^{2} + 3 y^{2} + 8 x y + y^{3} \end{aligned}$

:x2-2xy+y2)+(2y2-3x2)+(10xy+3)+(10x3) 类似术语组:=(x2-3x2)+(y2+2y2)+(-2xy+10xy)+(-2xy+10xy)+y3简化:\\2x2+3y2+8xy+y3

To subtract one polynomial from another, add the opposite of each term of the polynomial you are subtracting.
::要将一个多数值从另一个多数值中减去,请加上您在减的多数值中每个术语的相反值。

Subtracting Polynomials
::减减多元数

a) Subtract $\begin{aligned} x^{3} - 3 x^{2} + 8 x + 12 \end{aligned}$ from $\begin{aligned} 4 x^{2} + 5 x - 9 \end{aligned}$
:a) 从4x2+5x-9减去x3-3x2+8x+12

$\begin{aligned} (4 x^{2} + 5 x - 9) - (x^{3} - 3 x^{2} + 8 x + 12) & = (4 x^{2} + 5 x - 9) + (- x^{3} + 3 x^{2} - 8 x - 12) \\ Group like terms: & = - x^{3} + (4 x^{2} + 3 x^{2}) + (5 x - 8 x) + (- 9 - 12) \\ Simplify: & = - x^{3} + 7 x^{2} - 3 x - 21 \end{aligned}$

:4x2+5x-9)-(x3-3x2+8x+12)=(4x2+5x-9)+(-x3+3x2-8x-8x-12)

b) Subtract $\begin{aligned} 5 b^{2} - 2 a^{2} \end{aligned}$ from $\begin{aligned} 4 a^{2} - 8 a b - 9 b^{2} \end{aligned}$
::b) 从4a2-8ab-9b2中减去5b2-2a2

$\begin{aligned} (4 a^{2} - 8 a b - 9 b^{2}) - (5 b^{2} - 2 a^{2}) & = (4 a^{2} - 8 a b - 9 b^{2}) + (- 5 b^{2} + 2 a^{2}) \\ Group like terms: & = (4 a^{2} + 2 a^{2}) + (- 9 b^{2} - 5 b^{2}) - 8 a b \\ Simplify: & = 6 a^{2} - 14 b^{2} - 8 a b \end{aligned}$

:4a2-8ab-9b2)-(5b2-2a2)=(4a2-8ab-9b2)+(5b2+2a2)

Note: An easy way to check your work after adding or subtracting polynomials is to substitute a convenient value in for the variable , and check that your answer and the problem both give the same value. For example, in part (b) above, if we let $\begin{aligned} a = 2 \end{aligned}$ and $\begin{aligned} b = 3 \end{aligned}$ , then we can check as follows:
::注意: 添加或减去多数值后检查您的工作的一个简单方法就是替换变量中的方便值, 并检查您的答案和问题都具有相同的价值。例如, 在上文(b)部分, 如果我们使用 a=2 和 b=3, 那么我们可以检查如下 :

$\begin{aligned} Given & Solution \\ (4 a^{2} - 8 a b - 9 b^{2}) - (5 b^{2} - 2 a^{2}) & 6 a^{2} - 14 b^{2} - 8 a b \\ (4 (2)^{2} - 8 (2) (3) - 9 (3)^{2}) - (5 (3)^{2} - 2 (2)^{2}) & 6 (2)^{2} - 14 (3)^{2} - 8 (2) (3) \\ (4 (4) - 8 (2) (3) - 9 (9)) - (5 (9) - 2 (4)) & 6 (4) - 14 (9) - 8 (2) (3) \\ (- 113) - 37 & 24 - 126 - 48 \\ - 150 & - 150 \end{aligned}$

::4a2-8ab-9b2-(5b2-2-2a2-22-2a2)6a2-14b2-8ab2-8(2)(3)-9(3)2-(5(3)2-2-2-2(2)2)6(2)-2-14(3)-2-8(2)(3)(3)-4(4)-8(2)-(9)-(5(9)-(9)-2(4))6(4)-(9)-8(2)(3)(3)-(113)-3724-126-48-150-150

Since both expressions evaluate to the same number when we substitute in arbitrary values for the variables, we can be reasonably sure that our answer is correct.
::由于当我们用任意值取代变量时,这两个表达方式评价的数值相同,因此我们可以合理地确信我们的答案是正确的。

Note: When you use this method, do not choose 0 or 1 for checking since these can lead to common problems.
::注意:使用此方法时,不要选择 0 或 1 进行检查,因为这些方法可能导致常见问题。

Problem Solving Using Addition or Subtraction of Polynomials
::使用多面体加减法解决问题

One way we can use polynomials is to find the area of a geometric figure.
::我们可以使用多种数学的方法之一就是找到几何数字的面积。

Writing a Polynomial
::写作多面体

Write a polynomial that represents the area of each figure shown.
::写入一个代表显示的每个图的区域的多数值。

a)
::a) (a)

This shape is formed by two squares and two rectangles.
::这个形状由两个方形和两个矩形组成。

$\begin{aligned} The blue square has area y \times y & = y^{2} . \\ The yellow square has area x \times x & = x^{2} . \\ The pink rectangles each have area x \times y & = x y . \end{aligned}$

::蓝色方块的面积为 yxy=y2. 黄色方块的面积为 xxx=x2. 粉红色矩形的面积为 xxy=xy。

To find the total area of the figure we add all the separate areas:
::为了找到数字的总面积,我们加上所有单独的区域:

$\begin{aligned} T o t a l a r e a & = y^{2} + x^{2} + x y + x y \\ = y^{2} + x^{2} + 2 x y \end{aligned}$

::总面积=y2+x2+x2+xy+xy=y2+x2+2xy

b)
:b) b)

This shape is formed by two squares and one rectangle.
::这个形状由两个方形和一个矩形组成。

$\begin{aligned} The yellow squares each have area a \times a & = a^{2} . \\ The orange rectangle has area 2 a \times b & = 2 a b . \end{aligned}$

::黄色方块各有面积axa=a2. 橙色矩形有面积 2axb=2ab。

To find the total area of the figure we add all the separate areas:
::为了找到数字的总面积,我们加上所有单独的区域:

$\begin{aligned} T o t a l a r e a & = a^{2} + a^{2} + 2 a b \\ = 2 a^{2} + 2 a b \end{aligned}$

::总面积=a2+a2+2ab=2a2+2ab

c)
:c) c)

To find the area of the green region we find the area of the big square and subtract the area of the little square.
::为了找到绿区域,我们找到大广场地区,减去小广场地区。

$\begin{aligned} The big square has area : y \times y & = y^{2} . \\ The little square has area : x \times x & = x^{2} . \\ A r e a o f t h e g r e e n r e g i o n & = y^{2} - x^{2} \end{aligned}$

::大方块有面积:yxy=y2.小方块有面积:xxx=x2.绿区域区域=y2-x2

d)
::d) 与(d)

To find the area of the figure we can find the area of the big rectangle and add the areas of the pink squares.
::为了找到图中的区域,我们可以找到大矩形的区域,并添加粉红色方形的区域。

$\begin{aligned} The pink squares each have area a \times a & = a^{2} . \\ The blue rectangle has area 3 a \times a & = 3 a^{2} . \end{aligned}$

::粉红色方块各有面积axa=a2. 蓝色矩形有面积3axa=3a2。

To find the total area of the figure we add all the separate areas:
::为了找到数字的总面积,我们加上所有单独的区域:

$\begin{aligned} T o t a l a r e a = a^{2} + a^{2} + a^{2} + 3 a^{2} = 6 a^{2} \end{aligned}$

::总面积=a2+a2+a2+a2+3a2=6a2

Another way to find this area is to find the area of the big square and subtract the areas of the three yellow squares:
::找到这个区域的另一个方法是找到大广场的面积, 减去三个黄色广场的面积:

$\begin{aligned} The big square has area 3 a \times 3 a & = 9 a^{2} . \\ The yellow squares each have area a \times a & = a^{2} . \end{aligned}$

::大广场的面积为3ax3a=9a2.黄广场的面积分别为axa=a2。

To find the total area of the figure we subtract:
::为了找到数字的总面积,我们减去:

$\begin{aligned} A r e a & = 9 a^{2} - (a^{2} + a^{2} + a^{2}) \\ = 9 a^{2} - 3 a^{2} \\ = 6 a^{2} \end{aligned}$

::区域=9a2-(a2+a2+a2)=9a2-3a2=6a2

Example
::示例示例示例示例

Example 1
::例1

Subtract $\begin{aligned} 4 t^{2} + 7 t^{3} - 3 t - 5 \end{aligned}$ from $\begin{aligned} 6 t + 3 - 5 t^{3} + 9 t^{2} \end{aligned}$ .
::从 6t+3 - 5t3+9t2 中减4t2+7t3 - 3t- 5。

When subtracting polynomials, we have to remember to subtract each term. If the term is already negative, subtracting a negative term is the same thing as adding:
::在减去多数值时, 我们必须记住要减去每个术语。如果该术语已经是负值, 减去负值与添加相同 :

$\begin{aligned} 6 t + 3 - 5 t^{3} + 9 t^{2} - (4 t^{2} + 7 t^{3} - 3 t - 5) & = \\ 6 t + 3 - 5 t^{3} + 9 t^{2} - (4 t^{2}) - (7 t^{3}) - (- 3 t) - (- 5) & = \\ 6 t + 3 - 5 t^{3} + 9 t^{2} - 4 t^{2} - 7 t^{3} + 3 t + 5 & = \\ (6 t + 3 t) + (3 + 5) + (- 5 t^{3} - 7 t^{3}) + (9 t^{2} - 4 t^{2}) & = \\ 9 t + 8 - 12 t^{3} + 5 t^{2} & = \\ - 12 t^{3} + 5 t^{2} + 9 t + 8 \end{aligned}$

::6t+3-5t3+5t3+5t3+9t2-(4t2+7-3-3t5)=6t+3-5t3+5t3+5t3+5t3+5t2+5T2+5T2+5T2+5(3+5)+5+5+5+5)+(5t3-7t3+7t3+7t3+7t3+9t2+8-12t3+5t2+123+5T2+12t3+5T2+9t+8+8

The final answer is in standard form .
::最后的答案是标准格式。

Review
::回顾

Add and simplify.
::添加和简化。

$\begin{aligned} (x + 8) + (- 3 x - 5) \end{aligned}$
:x+8)+(-3x-5)

$\begin{aligned} (- 2 x^{2} + 4 x - 12) + (7 x + x^{2}) \end{aligned}$
:-2x2+4x-12)+(7x+x2)

$\begin{aligned} (2 a^{2} b - 2 a + 9) + (5 a^{2} b - 4 b + 5) \end{aligned}$
:2a2b-2a+9)+(5a2b-4b+5)

$\begin{aligned} (6.9 a^{2} - 2.3 b^{2} + 2 a b) + (3.1 a - 2.5 b^{2} + b) \end{aligned}$
:6.9a2-2.3b2+2ab)+(3.1a-2.5b2+b)

$\begin{aligned} (\frac{3}{5} x^{2} - \frac{1}{4} x + 4) + (\frac{1}{10} x^{2} + \frac{1}{2} x - 2 \frac{1}{5}) \end{aligned}$
:35x2-14x+4)+(110x2+12x-215)

Subtract and simplify.
::减法和简化。

$\begin{aligned} (- t + 5 t^{2}) - (5 t^{2} + 2 t - 9) \end{aligned}$
:-t+5t2)-(5t2+2-9)

$\begin{aligned} (- y^{2} + 4 y - 5) - (5 y^{2} + 2 y + 7) \end{aligned}$
:y2+4y-5)-(5y2+2y+7)

$\begin{aligned} (- 5 m^{2} - m) - (3 m^{2} + 4 m - 5) \end{aligned}$
:-5m2-m)-(3m2+4m-5)

$\begin{aligned} (2 a^{2} b - 3 a b^{2} + 5 a^{2} b^{2}) - (2 a^{2} b^{2} + 4 a^{2} b - 5 b^{2}) \end{aligned}$
:2a2b-3ab2+5a2b2)-(2a2b2+4a2b-2)

$\begin{aligned} (3.5 x^{2} y - 6 x y + 4 x) - (1.2 x^{2} y - x y + 2 y - 3) \end{aligned}$
:3.5x2y-6xy+4x)-(1.2x2y-xy+2y-3)

Find the area of the following figures.
::查找下列数字的区域。

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。

Section outline

Addition and Subtraction of Polynomials ::聚合物加和减减

Adding Polynomials ::添加多面体

Subtracting Polynomials ::减 减 多元数

Problem Solving Using Addition or Subtraction of Polynomials ::使用多面体加减法解决问题

Writing a Polynomial ::写作多面体

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Addition and Subtraction of Polynomials
::聚合物加和减减

Adding Polynomials
::添加多面体

Subtracting Polynomials
::减减多元数

Problem Solving Using Addition or Subtraction of Polynomials
::使用多面体加减法解决问题

Writing a Polynomial
::写作多面体

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)