12.6 帕斯卡尔三角和二亚米亚扩展系数

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• 选择节Pascal三角和Binomials扩张中的系数

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  Pascal三角和Binomials扩张中的系数

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

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  Pascal's Triangle
  ::帕斯卡尔三角

  

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  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, $\begin{array}{r}2+1=3\end{array}$ and $\begin{array}{r}10+10=20\end{array}$ . 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。

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  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。

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  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。

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  Let's continue the triangle to determine the elements in the $\begin{array}{r}{9}^{th}\end{array}$ 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。

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  Following the pattern of adding adjacent elements to get the elements in the next row, we find hat the eighth row is: $\begin{array}{r}172135352171\end{array}$
  ::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

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  Now, continue the pattern again to find the $\begin{array}{r}{9}^{th}\end{array}$ row: $\begin{array}{r}18285670562881\end{array}$
  ::现在,继续这个图案 找到第九排: 1 8 28 56 70 56 28 8 1

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  Now, let's expand the binomial $\begin{array}{r}(a+b{)}^4\end{array}$ 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。

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

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

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  We can see that the powers of $\begin{array}{r}a\end{array}$ start with 4 (the degree of the binomial) and decrease by one each term while the powers of $\begin{array}{r}b\end{array}$ 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 $\begin{array}{r}14641\end{array}$ , 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。

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  Finally, let's use what was discovered in the previous problem  to expand $\begin{array}{r}(x+y{)}^6\end{array}$ .
  ::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。

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  The degree of this expansion is 6, so the powers of $\begin{array}{r}x\end{array}$ will begin with 6 and decrease by one each term until reaching 0 while the powers of $\begin{array}{r}y\end{array}$ 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:

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  $$\begin{array}{r}\underset{\_}{}{x}^6+\underset{\_}{}{x}^{5}y+\underset{\_}{}{x}^4{y}^2+\underset{\_}{}{x}^3{y}^3+\underset{\_}{}{x}^2{y}^4+\underset{\_}{}x{y}^5+\underset{\_}{}{y}^6\end{array}$$

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

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  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 $\begin{array}{r}{6}^{th}\end{array}$ degree binomial, so the coefficients will be found in the $\begin{array}{r}{7}^{th}\end{array}$ 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。

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

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

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  Examples
  ::实例

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  Example 1
  ::例1

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

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  To expand the binomial $\begin{array}{r}(x-2{)}^5\end{array}$ , 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 。

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  The degree of this expansion is 5, so the powers of $\begin{array}{r}x\end{array}$ will begin with 5 and decrease by one each term until reaching 0 while the powers of $\begin{array}{r}y\end{array}$ , which in this case is $\begin{array}{r}-2\end{array}$ , 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:

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  $$\begin{array}{r}\underset{\_}{}{x}^5+\underset{\_}{}{x}^{4}y+\underset{\_}{}{x}^3{y}^2+\underset{\_}{}{x}^2{y}^3+\underset{\_}{}x{y}^4+\underset{\_}{}{y}^5\end{array}$$

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

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  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 $\begin{array}{r}-2\end{array}$ .
  ::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 。

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

  
  ::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

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  Example 2
  ::例2

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  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。

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  We continued the triangle to find the $\begin{array}{r}{9}^{th}\end{array}$ row earlier  and determined it  to be: $\begin{array}{r}18285670562881\end{array}$
  ::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

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  Subsequently, the $\begin{array}{r}{10}^{th}\end{array}$ row is: $\begin{array}{r}193684126126843691\end{array}$
  ::随后,第十排为: 1 9 36 84 126 126 84 36 9 1

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  Example 3
  ::例3

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

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

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

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  Example 4
  ::例4

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

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

  
  :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)

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  Review
  ::回顾

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  1. 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。
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  2. 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。

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  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。

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  3. $\begin{array}{r}(x-6{)}^4\end{array}$
     :x-6)4
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  4. $\begin{array}{r}(2x+5{)}^6\end{array}$
     :2x+5)6
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  5. $\begin{array}{r}(3-x{)}^7\end{array}$
     :3-x)7
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  6. $\begin{array}{r}(x^2-2{)}^3\end{array}$
     :x2-2)3
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  7. $\begin{array}{r}(x+4{)}^5\end{array}$
     :x+4)5
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  8. $\begin{array}{r}(2-x^3{)}^4\end{array}$
     :2-十三)4
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  9. $\begin{array}{r}(a-b{)}^6\end{array}$
     :a-b)6
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  10. $\begin{array}{r}(x+1{)}^{10}\end{array}$
      :x+1)10

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  Review (Answers)
  ::回顾(答复)

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  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 。