Section: Pascal三角和Binomials扩张中的系数 | 12.6 帕斯卡尔三角和二亚米亚扩展系数

Section outline

Without multiplying $\begin{aligned} (x - 2) \end{aligned}$ by itself five times, how could you expand $\begin{aligned} (x - 2)^{5} ? \end{aligned}$
::不自行乘(x-2)五倍,你怎么扩大(x-2)5?

Pascal's Triangle
::帕斯卡尔三角

Each row begins and ends with a one. Each “interior” value in each row is the sum of the two numbers above it. For example, $\begin{aligned} 2 + 1 = 3 \end{aligned}$ and $\begin{aligned} 10 + 10 = 20 \end{aligned}$ . This pattern produces the symmetry in the triangle.
::每个行以一开头和结尾。每行的每个“内部”值是上方两个数字的和。例如, 2+1=3 和 10+10=20。此图示在三角形中产生对称。

Another pattern that can be observed is that the row number is equal to the number of elements in that row. Row 1, for example has 1 element, 1. Row 2 has 2 elements, 1 and 1. Row 3 has 3 elements, 1, 2 and 1.
::另一个可以观察到的模式是,行号等于该行的元素数量。例如,行1有1个元素,行1有2个元素,行1有1个元素,行3有3个元素,1个元素,1个元素,2个元素,1个元素,1个元素,3个元素,1个元素,2个元素,1个元素,1个元素,1个元素,1个元素,3个元素,1个元素,2个元素,1个元素,1个元素,1个元素,1个元素,1个元素。

A third pattern is that the second element in the row is equal to one less than the row number. For example, in row 5 we have 1, 4, 6, 4 and 1.
::第三个模式是,行中的第二个元素等于行号少一个元素。例如,在第5行,我们有1、4、6、4和1个元素。

Let's continue the triangle to determine the elements in the $\begin{aligned} 9^{t h} \end{aligned}$ row of Pascal’s Triangle.
::让我们继续三角形来决定帕斯卡尔三角形第九排的元素。

Following the pattern of adding adjacent elements to get the elements in the next row, we find hat the eighth row is: $\begin{aligned} 1 7 21 35 35 21 7 1 \end{aligned}$
::按照在下一行添加相邻元素以获取元素的模式,我们发现第八排的帽子是:1 7 21 35 35 35 35 21 7 1

Now, continue the pattern again to find the $\begin{aligned} 9^{t h} \end{aligned}$ row: $\begin{aligned} 1 8 28 56 70 56 28 8 1 \end{aligned}$
::现在,继续这个图案找到第九排: 1 8 28 56 70 56 28 8 1

Now, let's expand the binomial $\begin{aligned} (a + b)^{4} \end{aligned}$ and discuss the pattern within the exponents of each variable as well as the pattern found in the coefficients of each term.
::现在,让我们扩展二进制(a+b) 4, 讨论每个变量的指数内的模式以及每个术语的系数中发现的模式。

$\begin{aligned} (a + b) (a + b) (a + b) (a + b) \\ (a^{2} + 2 a b + b^{2}) (a^{2} + 2 a b + b^{2}) \\ a^{4} + 2 a^{3} b + a^{2} & b^{2} + 2 a^{3} b + 4 a^{2} b^{2} + 2 a b^{3} + a^{2} b^{2} + 2 a b^{3} + b^{4} \\ a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4} \end{aligned}$

:a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a2) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a)+ (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a) (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)+ (a)

Take two binomials at a time and square them using $\begin{aligned} (a + b)^{2} = a^{2} + 2 a b + b^{2} \end{aligned}$
::使用 (a+b) 2=a2+2ab+b2, 一次取取两个二进制, 平方

Next, distribute each term in the first trinomial over each term in the second trinomial and collect like terms.
::接下来,在第一个三学期中将每个学期分配到第二个三学期,并收集同样的学期。

We can see that the powers of $\begin{aligned} a \end{aligned}$ start with 4 (the degree of the binomial) and decrease by one each term while the powers of $\begin{aligned} b \end{aligned}$ start with zero and increase by one each term. The number of terms is 5 which is one more than the degree of the binomial. The coefficients of the terms are $\begin{aligned} 1 4 6 4 1 \end{aligned}$ , the elements of row 5 in Pascal’s Triangle.
::我们可以看到,从4个开始(二进制程度)到每学期减少一个学期,B学期减少一个学期,B学期减少一个学期,B学期增加一个学期。学期数是5个学期数,比二进制程度多一个学期数。学期的系数是14 6 4 1,是帕斯卡尔三角第5行的元素。

Finally, let's use what was discovered in the previous problem to expand $\begin{aligned} (x + y)^{6} \end{aligned}$ .
::最后,让我们利用在前一个问题中发现的东西来扩展 (x+y) 6。

The degree of this expansion is 6, so the powers of $\begin{aligned} x \end{aligned}$ will begin with 6 and decrease by one each term until reaching 0 while the powers of $\begin{aligned} y \end{aligned}$ will begin with zero and increase by one each term until reaching 6. We can write the variables in the expansion (leaving space for the coefficients) as follows:
::扩展程度为6, 因此x的权力将从6开始,每个任期减少一个,直到达到0, 而y的权力从零开始,每个任期增加一个,直到达到6, 我们可以将扩展中的变量(系数的留置空间)写如下:

$\begin{aligned} \underline{} x^{6} + \underline{} x^{5} y + \underline{} x^{4} y^{2} + \underline{} x^{3} y^{3} + \underline{} x^{2} y^{4} + \underline{} x y^{5} + \underline{} y^{6} \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}==========================================================================================================

In the previous problem we observed that the coefficients for a fourth degree binomial were found in the fifth row of Pascal’s Triangle. Here we have a $\begin{aligned} 6^{t h} \end{aligned}$ degree binomial, so the coefficients will be found in the $\begin{aligned} 7^{t h} \end{aligned}$ row of Pascal’s Triangle. Now we can fill in the blanks with the correct coefficients.
::在上一个问题中,我们观察到,四度二进制系数出现在帕斯卡尔三角形第五行。我们这里有六度二进制系数,因此这些系数将出现在帕斯卡尔三角形第七行。现在我们可以用正确的系数填充空白。

$\begin{aligned} x^{6} + 6 x^{5} y + 15 x^{4} y^{2} + 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 x y^{5} + y^{6} \end{aligned}$

::x6+6x5y+15x4y2+20x3y3+15x2y4+6x5+y6

Examples
::实例

Example 1
::例1

Earlier, you were asked to expand the binomial $\begin{aligned} (x - 2)^{5} \end{aligned}$ .
::早些时候,有人要求你扩大二进制(x-2)5。

To expand the binomial $\begin{aligned} (x - 2)^{5} \end{aligned}$ , you could use Pascal's Triangle.
::要扩展二进制( x-2 5) , 您可以使用帕斯卡尔的三角形。

The degree of this expansion is 5, so the powers of $\begin{aligned} x \end{aligned}$ will begin with 5 and decrease by one each term until reaching 0 while the powers of $\begin{aligned} y \end{aligned}$ , which in this case is $\begin{aligned} - 2 \end{aligned}$ , will begin with zero and increase by one each term until reaching 5. We can write the variables in the expansion (leaving space for the coefficients) as follows:
::扩展程度为5,因此x的权力将从5开始,每个任期减少一个,直到达到0,y的权力(在本案中为-2)将从零开始,每个任期增加一个,直到达到5。我们可以将扩展中的变量(系数的留置空间)写如下:

$\begin{aligned} \underline{} x^{5} + \underline{} x^{4} y + \underline{} x^{3} y^{2} + \underline{} x^{2} y^{3} + \underline{} x y^{4} + \underline{} y^{5} \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

The coefficients for a fifth degree binomial can be found in the sixth row of Pascal’s Triangle. Now we can fill in the blanks with the correct coefficients, replacing y with $\begin{aligned} - 2 \end{aligned}$ .
::第五度二进制系数可以在帕斯卡尔三角形第六行找到。现在我们可以用正确的系数填空空, 替换 y 。

$\begin{aligned} 1 \cdot x^{5} + 5 x^{4} (- 2) + 10 x^{3} (- 2)^{2} + 10 x^{2} (- 2)^{3} + 5 x (- 2)^{4} + 1 \cdot (- 2)^{5} \\ x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 80 x - 32 \end{aligned}$

::1x5+5x4(-2-2)+10x3(-2-2)+10x3(-2)2+10x2(-2)3+5x(-2)4+1(-2)4+1(-2)5x5-10x4+40x3-80x2+80x-32

Example 2
::例2

Write out the elements in row 10 of Pascal’s Triangle.
::将元素写在帕斯卡尔三角的第10行。

We continued the triangle to find the $\begin{aligned} 9^{t h} \end{aligned}$ row earlier and determined it to be: $\begin{aligned} 1 8 28 56 70 56 28 8 1 \end{aligned}$
::我们继续三角形之前找到了第九排确定为: 1 8 28 56 70 56 28 8 1

Subsequently, the $\begin{aligned} 10^{t h} \end{aligned}$ row is: $\begin{aligned} 1 9 36 84 126 126 84 36 9 1 \end{aligned}$
::随后,第十排为: 1 9 36 84 126 126 84 36 9 1

Example 3
::例3

Expand $\begin{aligned} (a + 4)^{3} \end{aligned}$ .
::展开(a+4) 3.

$\begin{aligned} a^{3} & + 3 a^{2} (4) + 3 a (4)^{2} + (4)^{3} \\ a^{3} + 12 a^{2} + 48 a + 64 \end{aligned}$

::a3+3a2(4)+3a(4)+4a(4)2+(4)3a3+12a2+48a+64

Example 4
::例4

Write out the coefficients in the expansion of $\begin{aligned} (2 x - 3)^{4} \end{aligned}$ .
::写出扩大的系数(2x-3)4。

$\begin{aligned} (2 x)^{4} + 4 & (2 x)^{3} (- 3) + 6 (2 x)^{2} (- 3)^{2} + 4 (2 x) (- 3)^{3} + (- 3)^{4} \\ 16 x^{4} - 96 x^{3} + 216 x^{2} - 216 x + 81 \end{aligned}$

:2x)4+4+4(2x)3(3)+6(2x)(2-3)+4(2x)(3)+4(2x)(3)3+3(3)+(3)416x4-96x3+216x2-216x2+81)

Review
::回顾

Write out the elements in row 7 of Pascal’s Triangle.
::将元素写在帕斯卡尔三角形第7行。

Write out the elements in row 13 of Pascal’s Triangle.
::将元素写在帕斯卡尔三角的第13行。

Use Pascal’s Triangle to expand the following binomials.
::使用帕斯卡尔三角以扩展以下二进制。

$\begin{aligned} (x - 6)^{4} \end{aligned}$
:x-6)4

$\begin{aligned} (2 x + 5)^{6} \end{aligned}$
:2x+5)6

$\begin{aligned} (3 - x)^{7} \end{aligned}$
:3-x)7

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

$\begin{aligned} (x + 4)^{5} \end{aligned}$
:x+4)5

$\begin{aligned} (2 - x^{3})^{4} \end{aligned}$
:2-十三)4

$\begin{aligned} (a - b)^{6} \end{aligned}$
:a-b)6

$\begin{aligned} (x + 1)^{10} \end{aligned}$
:x+1)10

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

Pascal's Triangle ::帕斯卡尔三角

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Pascal's Triangle
::帕斯卡尔三角

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)