Course: 9.4 双义体对多义体的乘法

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二进制乘多式乘法
Multiplication of Polynomials by Binomials
::二进制乘多式乘法

Let’s start by multiplying two binomials together. A binomial is a polynomial with two terms , so a product of two binomials will take the form $\begin{aligned} (a + b) (c + d) \end{aligned}$ .
::让我们首先将两个二进制相乘。一个二进制是一个包含两个条件的多元性,因此两个二进制的产物将呈现形式(a+b(c)+d ) 。

We can still use the here if we do it cleverly. First, let’s think of the first set of " data-term="Parentheses" role="term" tabindex="0"> parentheses as one term . The Distributive Property says that we can multiply that term by $\begin{aligned} c \end{aligned}$ , multiply it by $\begin{aligned} d \end{aligned}$ , and then add those two products together: $\begin{aligned} (a + b) (c + d) = (a + b) \cdot c + (a + b) \cdot d \end{aligned}$ .
::如果我们聪明地这样做,我们仍然可以使用这里。首先,让我们把第一组括号当作一个术语。分配属性表示我们可以将这一术语乘以 c, 乘以 d, 然后将这两个产品加在一起a+b)(c+d) = (a+b) c+(a+b) +(a+b) +(a+b) +(a+b) d。

We can rewrite this expression as $\begin{aligned} c (a + b) + d (a + b) \end{aligned}$ . Now let’s look at each half separately. We can apply the distributive property again to each set of parentheses in turn, and that gives us $\begin{aligned} c (a + b) + d (a + b) = c a + c b + d a + d b . \end{aligned}$
::我们可以重写此表达式为 c( a+b) +d( a+b) +d( a+b) 。现在让我们分别查看每半个。我们可以将分配属性再次应用到每组括号中, 这样我们就可以使用 c( a+b) +d( a+b) =ca+cb+da+db 。

What you should notice is that when multiplying any two , every term in one polynomial is multiplied by every term in the other polynomial .
::您应该注意的是,当乘以任何两个时,一个多义中每个词的乘以另一个多义中每个词的乘以。

Multiplying and Simplifying
::乘和简化

1. Multiply and simplify: $\begin{aligned} (2 x + 1) (x + 3) \end{aligned}$
::1. 乘数和简化2x+1)(x+3)

We must multiply each term in the first polynomial by each term in the second polynomial. Let’s try to be systematic to make sure that we get all the products.
::我们必须把第一个多民族术语中的每个术语乘以第二个多民族术语中的每个术语。让我们尝试系统化地确保我们获得所有产品。

First, multiply the first term in the first set of parentheses by all the terms in the second set of parentheses.
::首先,将第一组括号中的第一个任期乘以第二组括号中的所有用语。

Now we’re done with the first term. Next we multiply the second term in the first parentheses by all terms in the second parentheses and add them to the previous terms.
::现在,我们完成了第一个任期。接下来,我们将第一个括号中第二个任期乘以第二个括号中的所有术语,然后将其添加到前几个任期中。

Now we’re done with the multiplication and we can simplify:
::现在,我们完成了乘法的乘法,我们可以简化:

$\begin{aligned} (2 x) (x) + (2 x) (3) + (1) (x) + (1) (3) = 2 x^{2} + 6 x + x + 3 = 2 x^{2} + 7 x + 3 \end{aligned}$
:2x)+(2x)+(2x)(3)+(1)(x)+(1x)+(1)(3)=2x2+6x+x+3=2x2+7x+3)

This way of multiplying polynomials is called in-line multiplication or horizontal multiplication. Another method for multiplying polynomials is to use vertical multiplication, similar to the vertical multiplication you learned with regular numbers.
::倍增多数值的这种方式被称为线性倍增或水平倍增。倍增多数值的另一种方法是使用垂直倍增, 类似于您用普通数字所学的垂直倍增。

2. Multiply and simplify:
::2. 乘以和简化:

a) $\begin{aligned} (4 x - 5) (x - 20) \end{aligned}$
:a) (4x-5(x-20))

With horizontal multiplication this would be
::随着水平乘法的横向乘法

$\begin{aligned} (4 x - 5) (x - 20) & = (4 x) (x) + (4 x) (- 20) + (- 5) (x) + (- 5) (- 20) \\ = 4 x^{2} - 80 x - 5 x + 100 \\ = 4 x^{2} - 85 x + 100 \end{aligned}$

:4x-5(x-20)=(4x)(x)+(4x)(4x)(-20)+(-5)(x)+(-5)(5)(-20)+(-5)(5)(20)=4x2-80x-5x+100=4x2-85x+100)

To do vertical multiplication instead, we arrange the polynomials on top of each other with like terms in the same columns:
::为了进行垂直乘法,我们用同一列中的类似术语,在彼此的顶部安排多式乘法:

$\begin{aligned} 4 x - 5 \\ \underline{x - 20} \\ - 80 x + 100 \\ \underline{4 x^{2} - 5 x} \\ 4 x^{2} - 85 x + 100 \end{aligned}$

::4 - 5x - 20_- 80x+1004x2 - 5x_ 4x2 - 85x+100

Both techniques result in the same answer: $\begin{aligned} 4 x^{2} - 85 x + 100 \end{aligned}$ . We’ll use vertical multiplication for the other problems.
::这两种技术的结果都是相同的答案:4x2-85x+100。我们用垂直乘法解决其他问题。

b) $\begin{aligned} (3 x - 2) (3 x + 2) \end{aligned}$
:3x-2(3x+2))

$\begin{aligned} 3 x - 2 \\ \underline{3 x + 2} \\ 6 x - 4 \\ \underline{9 x^{2} - 6 x} \\ 9 x^{2} + 0 x - 4 \end{aligned}$

::3 - 23x+2_ 6x-49x2-6x_ 9x2+0x-4

The answer is $\begin{aligned} 9 x^{2} - 4 \end{aligned}$ .
::答案是 9x2 -4。

c) $\begin{aligned} (3 x^{2} + 2 x - 5) (2 x - 3) \end{aligned}$
:c) (3x2+2x-5)(2x-3)

It’s better to place the smaller polynomial on the bottom:
::最好把小多面体放在底部:

$\begin{aligned} 3 x^{2} + 2 x - 5 \\ \underline{2 x - 3} \\ - 9 x^{2} - 6 x + 15 \\ \underline{6 x^{3} + 4 x^{2} - 10 x} \\ 6 x^{3} - 5 x^{2} - 16 x + 15 \end{aligned}$

::3x2+2x-52x-3_-9x2-6x+156x3+4x2-10x_6x3-5x2-16x+15

The answer is $\begin{aligned} 6 x^{3} - 5 x^{2} - 16 x + 15 \end{aligned}$ .
::答案是 6x3 - 5x2 - 16x+15。

d) $\begin{aligned} (x^{2} - 9) (4 x^{4} + 5 x^{2} - 2) \end{aligned}$
:x2-9)(4x4+5x2-2)

Set up the multiplication vertically and leave gaps for missing powers of $\begin{aligned} x \end{aligned}$ :
::垂直设置乘法并留下X 缺失功率的空白 :

$\begin{aligned} 4 x^{4} + 5 x^{2} - 2 \\ \underline{x^{2} - 9} \\ - 36 x^{4} - 45 x^{2} + 18 \\ \underline{4 x^{6} + 5 x^{4} - 2 x^{2}} \\ 4 x^{6} - 31 x^{4} - 47 x^{2} + 18 \end{aligned}$

::4x4+5x2-2-2x2-9_-36x4-45x2+184x6+5x4-2x2_4x6-314-47x2+18

The answer is $\begin{aligned} 4 x^{6} - 31 x^{4} - 47 x^{2} + 18 \end{aligned}$ .
::答案是4x6-31x4-47x2+18。

Solve Real-World Problems Using Multiplication of Polynomials
::利用多面体乘法解决现实世界问题

In this section, we’ll see how multiplication of polynomials is applied to finding the areas and volumes of geometric shapes.
::本节将说明多元形的倍增是如何应用于寻找几何形状的面积和体积的。

Real-World Application: Finding Areas and Volumes
::现实世界应用:发现地区和数量

a) Find the areas of the figure:
:a) 找出图中所涉领域:

We use the formula for the area of a rectangle: $\begin{aligned} Area = length \times width \end{aligned}$ .
::我们对矩形区域使用公式:区域=长xwidth。

For the big rectangle:
::对于大矩形:

$\begin{aligned} Length & = b + 3, Width = b + 2 \\ Area & = (b + 3) (b + 2) \\ = b^{2} + 2 b + 3 b + 6 \\ = b^{2} + 5 b + 6 \end{aligned}$

::长度=b+3,宽度=b+2Area=(b+3)(b+2)=b2+2b+3b+6=b2+5b+6

b) Find the volumes of the figure:
:b) 查找图中的数字数量:

The volume of this shape = (area of the base)(height).
::此形状的大小 =( 底部区域)( 高度) 。

$\begin{aligned} Area of the base & = x (x + 2) \\ = x^{2} + 2 x \\ Height & = 2 x + 1 \\ Volume & = (x^{2} + 2 x) (2 x + 1) \\ = 2 x^{3} + x^{2} + 4 x^{2} + 2 x \\ = 2 x^{3} + 5 x^{2} + 2 x \end{aligned}$

::基数面积=xx( x+2) =x2+2x58=2x1 Volume=( x2+2x)( 2x+1) =2x3+x2+4x2+2+2x2+2x=2x3+5x2+2x

Examples
::实例

Example 1
::例1

Find the areas of the figure:
::查找图中的区域 :

We could add up the areas of the blue and orange rectangles, but it’s easier to just find the area of the whole big rectangle and subtract the area of the yellow rectangle.
::我们可以将蓝色和橙色矩形的面积加在一起, 但只要找到整个大矩形的面积, 并减去黄色矩形的面积就更容易了。

$\begin{aligned} Area of big rectangle & = 20 (12) = 240 \\ Area of yellow rectangle & = (12 - x) (20 - 2 x) \\ = 240 - 24 x - 20 x + 2 x^{2} \\ = 240 - 44 x + 2 x^{2} \\ = 2 x^{2} - 44 x + 240 \end{aligned}$

::大矩形面积=20(12)=240 黄矩形区域=(12-x)(20-2x)=240-24x-20x+2x2=240-44x+2x2=2x2-44x2=2x2-44x+240

The desired area is the difference between the two:
::理想的面积是两者之间的差别:

$\begin{aligned} Area & = 240 - (2 x^{2} - 44 x + 240) \\ = 240 + (- 2 x^{2} + 44 x - 240) \\ = 240 - 2 x^{2} + 44 x - 240 \\ = - 2 x^{2} + 44 x \end{aligned}$

::区域=240 - (2x2- 44x+240)=240+(-2x2+44x- 240)=240-2x2+442- 44xx- 240x

Example 2
::例2

Find the volumes of the figure:
::查找图的音量 :

The volume of this shape = (area of the base)(height).
::此形状的大小 =( 底部区域)( 高度) 。

$\begin{aligned} Area of the base & = (4 a - 3) (2 a + 1) \\ = 8 a^{2} + 4 a - 6 a - 3 \\ = 8 a^{2} - 2 a - 3 \\ Height & = a + 4 \\ Volume & = (8 a^{2} - 2 a - 3) (a + 4) \end{aligned}$

:4a-3)(2a+1)=8a2+4a-6a-3=8a2-2a-3H8=a+4Volume=(8a2-2a-3)(a+4)

Let’s multiply using the vertical method:
::让我们使用垂直方法乘以 :

$\begin{aligned} 8 a^{2} - 2 a - 3 \\ \underline{a + 4} \\ 32 a^{2} - 8 a - 12 \\ \underline{8 a^{3} - 2 a^{2} - 3 a} \\ 8 a^{3} + 30 a^{2} - 11 a - 12 \end{aligned}$

::8a2-2a - 3a - 3a+4_ 32a2 - 8a - 128a3 - 2a2 - 3a8a3+ 30a2 - 11a - 12

The volume is $\begin{aligned} 8 a^{3} + 30 a^{2} - 11 a - 12 \end{aligned}$ .
::卷号为8a3+30a2-11a-12。

Review
::回顾

Multiply and simplify.
::乘数和简化。

$\begin{aligned} (x - 3) (x + 2) \end{aligned}$
:x-3(x+2))

$\begin{aligned} (a + b) (a - 5) \end{aligned}$
:a+b(a-5)

$\begin{aligned} (x + 2) (x^{2} - 3) \end{aligned}$
:x+2)(x2-3)

$\begin{aligned} (a^{2} + 2) (3 a^{2} - 4) \end{aligned}$
:a2+2)(3a2-4)

$\begin{aligned} (7 x - 2) (9 x - 5) \end{aligned}$
:7x-2(9x-5))

$\begin{aligned} (2 x - 1) (2 x^{2} - x + 3) \end{aligned}$
:2x-1)(2x2-x+3)

$\begin{aligned} (3 x + 2) (9 x^{2} - 6 x + 4) \end{aligned}$
:3x+2)(9x2-6x+4)

$\begin{aligned} (a^{2} + 2 a - 3) (a^{2} - 3 a + 4) \end{aligned}$
:a2+2a-3(a2-3a+4)

$\begin{aligned} 3 (x - 5) (2 x + 7) \end{aligned}$
::3(x-5)(2x+7)

$\begin{aligned} 5 x (x + 4) (2 x - 3) \end{aligned}$
::5x(xx+4)(2x-3)

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

Find the volumes of the following figures.
::查找下列数字的卷数。

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

General

二进制乘多式乘法

Multiplication of Polynomials by Binomials ::二进制乘多式乘法

Multiplying and Simplifying ::乘和简化

Solve Real-World Problems Using Multiplication of Polynomials ::利用多面体乘法解决现实世界问题

Real-World Application: Finding Areas and Volumes ::现实世界应用:发现地区和数量

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Multiplication of Polynomials by Binomials
::二进制乘多式乘法

Multiplying and Simplifying
::乘和简化

Solve Real-World Problems Using Multiplication of Polynomials
::利用多面体乘法解决现实世界问题

Real-World Application: Finding Areas and Volumes
::现实世界应用:发现地区和数量

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)