每天清晨，当第一缕阳光洒在湖面上，一个身影便会出现在湖心小岛上。她坐在一块大石头上，周围被茂盛的植物环绕，安静地沉浸在数学的世界中。
这个姑娘叫小悦，她的故事在这个美丽的湖心小岛上展开。每天早晨，她都会提前来到湖边，仔细观察水下的植物，然后抽出时间来钻研一元x次方程。她身上的气息混合着湖水的清新和植物的芬芳，形成一种独特的味道，让人感到宁静与祥和。
然而，一元x次方程的展开对于小悦来说，并不是一件容易的事。这个看似简单的数学问题，却困扰了她许久。然而，小悦并没有向困难低头，她坚信，只要努力，就一定能够找到解决的方法。
在这座小岛上，小悦度过了无数个早晨。她反复琢磨着方程的特点，尝试寻找解法。有时候，她会陷入深深的思考，甚至忘记时间；有时候，她会突然灵光一闪，兴奋地写下展开式的公式。每一个早晨，小悦都在进步，她的眼中闪耀着对知识的渴望和对梦想的坚定。
终于有一天，通过前面的积累，小悦灵光一闪，意识到她可以通过将一元x次方程的每一项分别展开，然后再将这些展开式合并起来，得到一元x次方程的展开式。于是她拿起笔和纸，开始耐心地展开每一项。首先，她展开了一元x次方程中的常数项，接着展开了一次项、二次项、三次项……，最后将所有展开式合并起来，得到了一元x次方程的展开式。小悦看着自己长期努力得来的成果，激动得热泪盈眶。
她无法掩饰内心的喜悦，兴奋地在湖边跳跃着。湖面上的波纹在阳光的照射下闪着金光，似乎在为她的成功欢呼。那一刻，小悦觉得自己仿佛成为了湖水的一部分，与周围的环境融为一体。
随着时间的推移，小悦在岛上的生活也变得更加丰富多彩。她开始尝试将数学知识应用到日常生活中，在烹饪时运用几何学来切蛋糕，或者在散步时用代数知识来计算最短路径问题。这些小小的尝试让小悦意识到，知识不仅仅是为了考试和学术，它更是一种工具，可以帮助她更好地生活。
这个美丽的湖心小岛成为了小悦成长的见证。她在知识的海洋中探索，用数学来解读自然界的奥秘。清晨的阳光照耀在她的书桌上，给她带来温暖和勇气。傍晚时分，当夕阳洒在湖面上，小悦坐在窗前，静静地看着湖面的金辉渐渐消失在暮色中。
小悦面临的一元多次方程的展开式问题如下，她是如何处理呢：
输入一个带有一个单字符变量的表达式，并将其展开。表达式的形式为（ax+b）^n，其中a和b是整数，可以是正的，也可以是负的，x是任何单字符变量，n是自然数。如果a=1，则变量前面不会放置任何系数。如果a=-1，则变量前面将放一个“-”。
展开后的表达式应以字符串形式返回，格式为ax^b+cx^d+ex^f。。。其中a、c和e是项的系数，x是原始表达式中传递的原始一个字符变量，b、d和f是每个项中x的幂，并且是递减的。
如果项的系数为零，则不应包括该项。如果一个项的系数为1，则不应包括该系数。如果项的系数为-1，则只应包含“-”。如果项的幂为0，则只应包括系数。如果项的幂为1，则应排除插入符号和幂。
示例：
EdmSolution.Expand("(x+1)^2"); // returns "x^2+2x+1"
EdmSolution.Expand("(p-1)^3"); // returns "p^3-3p^2+3p-1"
EdmSolution.Expand("(2f+4)^6"); // returns "64f^6+768f^5+3840f^4+10240f^3+15360f^2+12288f+4096"
EdmSolution.Expand("(-2a-4)^0"); // returns "1"
EdmSolution.Expand("(-12t+43)^2"); // returns "144t^2-1032t+1849"
EdmSolution.Expand("(r+0)^203"); // returns "r^203"
EdmSolution.Expand("(-x-1)^2"); // returns "x^2+2x+1"
算法实现：
1 public class EdmSolution
2 {
3 // 定义一个只读的静态正则表达式对象，用于匹配表达式的模式
4 private readonly static Regex pattern = new Regex( @" ^\((-?\d*)(.)([-+]\d+)\)\^(\d+)$ " , RegexOptions.Compiled);
5
6 // 定义一个静态方法，用于展开给定的表达式
7 public static string Expand( string expr)
8 {
9 // 使用正则表达式匹配给定的表达式，并将匹配结果转换为字符串数组
10 var matches = pattern.Matches(expr).Cast<Match>().First().Groups.Cast<Group>().Skip( 1 ).Select(g => g.Value).ToArray();
11
12 // 解析匹配结果中的各个分组，并赋值给对应的变量
13 var a = matches[ 0 ].Length == 0 ? 1 : matches[ 0 ] == " - " ? - 1 : int .Parse(matches[ 0 ]);
14 var x = matches[ 1 ];
15 var b = int .Parse(matches[ 2 ]);
16 var n = int .Parse(matches[ 3 ]);
17
18 // 计算系数f的初始值，使用BigInteger类处理大整数
19 var f = new BigInteger(Math.Pow(a, n));
20
21 // 根据系数f的值确定常数c的值
22 var c = f == - 1 ? " - " : f == 1 ? "" : f.ToString();
23
24 // 处理特殊情况：指数为0或常数为0的情况
25 if (n == 0 ) return " 1 " ;
26 if (b == 0 ) return $ " {c}{x}{(n > 1) ? " ^ " : "" }{n} " ;
27
28 // 创建一个StringBuilder对象，用于存储展开后的表达式
29 var res = new StringBuilder();
30
31 // 循环展开表达式的每一项
32 for ( var i = 0 ; i <= n; i++ )
33 {
34 // 根据系数f的符号和当前项的位置，添加"+"或"-"符号
35 if (f > 0 && i > 0 ) res.Append( " + " );
36 if (f < 0 ) res.Append( " - " );
37
38 // 添加系数的绝对值，如果系数大于1或当前项不是第一项
39 if (i > 0 || f * f > 1 ) res.Append($ " {BigInteger.Abs(f)} " );
40
41 // 添加变量x，如果当前项不是最后一项
42 if (i < n) res.Append(x);
43
44 // 添加指数符号和指数值，如果当前项不是倒数第二项
45 if (i < n - 1 ) res.Append($ " ^{n - i} " );
46
47 // 更新系数f的值
48 f = f * (n - i) * b / a / (i + 1 );
49 }
50
51 // 将StringBuilder对象转换为字符串，并返回展开后的表达式
52 return res.ToString();
53 }
54 }
算法运行步骤：EdmSolution.Expand("(-5m+3)^4")
1. 匹配表达式：(-5m+3)^4
2. 使用正则表达式匹配给定的表达式，得到匹配结果：
- matches[0] = "-5"
- matches[1] = "m"
- matches[2] = "+3"
- matches[3] = "4"
3. 解析匹配结果中的各个分组：
- a = -5
- x = "m"
- b = 3
- n = 4
4. 计算系数f的初始值：f = (-5)^4 = 625
5. 根据系数f的值确定常数c的值：c = ""
6. 检查特殊情况：n = 4，不为0；b = 3，不为0
7. 创建StringBuilder对象res，用于存储展开后的表达式
8. 开始循环展开表达式的每一项：
- 第一项：i = 0
- f > 0，不添加"+"符号
- f * f > 1，添加系数的绝对值：625
- i < n，添加变量x："m"
- i < n - 1，添加指数符号和指数值："^4"
- 更新系数f的值：f = 625 * (4 - 0) * 3 / -5 / (0 + 1) = -1500
- 第二项：i = 1
- f < 0，添加"-"符号
- f * f > 1，添加系数的绝对值：1500
- i < n，添加变量x："m"
- i < n - 1，添加指数符号和指数值："^3"
- 更新系数f的值：f = -1500 * (4 - 1) * 3 / -5 / (1 + 1) = 1350
- 第三项：i = 2
- f < 0，添加"-"符号
- f * f > 1，添加系数的绝对值：1350
- i < n，添加变量x："m"
- i < n - 1，添加指数符号和指数值："^2"
- 更新系数f的值：f = 1350 * (4 - 2) * 3 / -5 / (2 + 1) = -540
- 第四项：i = 3
- f < 0，添加"-"符号
- f * f > 1，添加系数的绝对值：540
- i < n，添加变量x："m"
- i < n - 1，不添加指数符号和指数值
- 更新系数f的值：f = 540 * (4 - 3) * 3 / -5 / (3 + 1) = 81
- 第五项：i = 4
- f < 0，添加"-"符号
- f * f > 1，添加系数的绝对值：81
- i < n，不添加变量x
- i < n - 1，不添加指数符号和指数值
- 更新系数f的值：f = 81 * (4 - 4) * 3 / -5 / (4 + 1) = 0
9. 循环结束，返回StringBuilder对象res转换后的字符串："625m^4-1500m^3+1350m^2-540m+81"
10. 断言结果与期望值相等，测试通过
测试用例：
1 namespace Solution
2 {
3 using NUnit.Framework;
4 using System;
5 using System.Collections.Generic;
6 using System.Text;
7 using System.Text.RegularExpressions;
8
9 [TestFixture]
10 public class SolutionTest
11 {
12 [Test]
13 public void testBPositive()
14 {
15 Assert.AreEqual( " 1 " , EdmSolution.Expand( " (x+1)^0 " ));
16 Assert.AreEqual( " x+1 " , EdmSolution.Expand( " (x+1)^1 " ));
17 Assert.AreEqual( " x^2+2x+1 " , EdmSolution.Expand( " (x+1)^2 " ));
18 Assert.AreEqual( " x^3+3x^2+3x+1 " , EdmSolution.Expand( " (x+1)^3 " ));
19 Assert.AreEqual( " x^4+4x^3+6x^2+4x+1 " , EdmSolution.Expand( " (x+1)^4 " ));
20 Assert.AreEqual( " x^5+5x^4+10x^3+10x^2+5x+1 " , EdmSolution.Expand( " (x+1)^5 " ));
21 Assert.AreEqual( " 1 " , EdmSolution.Expand( " (x+2)^0 " ));
22 Assert.AreEqual( " x+2 " , EdmSolution.Expand( " (x+2)^1 " ));
23 Assert.AreEqual( " x^2+4x+4 " , EdmSolution.Expand( " (x+2)^2 " ));
24 Assert.AreEqual( " x^3+6x^2+12x+8 " , EdmSolution.Expand( " (x+2)^3 " ));
25 Assert.AreEqual( " x^4+8x^3+24x^2+32x+16 " , EdmSolution.Expand( " (x+2)^4 " ));
26 Assert.AreEqual( " x^5+10x^4+40x^3+80x^2+80x+32 " , EdmSolution.Expand( " (x+2)^5 " ));
27 Assert.AreEqual( " t^5+10t^4+40t^3+80t^2+80t+32 " , EdmSolution.Expand( " (t+2)^5 " ));
28 Assert.AreEqual( " y^15+75y^14+2625y^13+56875y^12+853125y^11+9384375y^10+78203125y^9+502734375y^8+2513671875y^7+9775390625y^6+29326171875y^5+66650390625y^4+111083984375y^3+128173828125y^2+91552734375y+30517578125 " , EdmSolution.Expand( " (y+5)^15 " ));
29 }
30
31 [Test]
32 public void testBNegative()
33 {
34 Assert.AreEqual( " 1 " , EdmSolution.Expand( " (x-1)^0 " ));
35 Assert.AreEqual( " x-1 " , EdmSolution.Expand( " (x-1)^1 " ));
36 Assert.AreEqual( " x^2-2x+1 " , EdmSolution.Expand( " (x-1)^2 " ));
37 Assert.AreEqual( " x^3-3x^2+3x-1 " , EdmSolution.Expand( " (x-1)^3 " ));
38 Assert.AreEqual( " x^4-4x^3+6x^2-4x+1 " , EdmSolution.Expand( " (x-1)^4 " ));
39 Assert.AreEqual( " x^5-5x^4+10x^3-10x^2+5x-1 " , EdmSolution.Expand( " (x-1)^5 " ));
40 Assert.AreEqual( " 1 " , EdmSolution.Expand( " (x-2)^0 " ));
41 Assert.AreEqual( " x-2 " , EdmSolution.Expand( " (x-2)^1 " ));
42 Assert.AreEqual( " x^2-4x+4 " , EdmSolution.Expand( " (x-2)^2 " ));
43 Assert.AreEqual( " x^3-6x^2+12x-8 " , EdmSolution.Expand( " (x-2)^3 " ));
44 Assert.AreEqual( " x^4-8x^3+24x^2-32x+16 " , EdmSolution.Expand( " (x-2)^4 " ));
45 Assert.AreEqual( " x^5-10x^4+40x^3-80x^2+80x-32 " , EdmSolution.Expand( " (x-2)^5 " ));
46 Assert.AreEqual( " t^5-10t^4+40t^3-80t^2+80t-32 " , EdmSolution.Expand( " (t-2)^5 " ));
47 Assert.AreEqual( " y^15-75y^14+2625y^13-56875y^12+853125y^11-9384375y^10+78203125y^9-502734375y^8+2513671875y^7-9775390625y^6+29326171875y^5-66650390625y^4+111083984375y^3-128173828125y^2+91552734375y-30517578125 " , EdmSolution.Expand( " (y-5)^15 " ));
48 }
49
50 [Test]
51 public void testAPositive()
52 {
53 Assert.AreEqual( " 625m^4+1500m^3+1350m^2+540m+81 " , EdmSolution.Expand( " (5m+3)^4 " ));
54 Assert.AreEqual( " 8x^3-36x^2+54x-27 " , EdmSolution.Expand( " (2x-3)^3 " ));
55 Assert.AreEqual( " 1 " , EdmSolution.Expand( " (7x-7)^0 " ));
56 Assert.AreEqual( " 35831808a^7+20901888a^6+5225472a^5+725760a^4+60480a^3+3024a^2+84a+1 " , EdmSolution.Expand( " (12a+1)^7 " ));
57 Assert.AreEqual( " 184528125x^5-123018750x^4+32805000x^3-4374000x^2+291600x-7776 " , EdmSolution.Expand( " (45x-6)^5 " ));
58 Assert.AreEqual( " 12c+1 " , EdmSolution.Expand( " (12c+1)^1 " ));
59 Assert.AreEqual( " 100000000x^4-4000000x^3+60000x^2-400x+1 " , EdmSolution.Expand( " (100x-1)^4 " ));
60 Assert.AreEqual( " 1000x^3+2400x^2+1920x+512 " , EdmSolution.Expand( " (10x+8)^3 " ));
61 Assert.AreEqual( " 128x^7-448x^6+672x^5-560x^4+280x^3-84x^2+14x-1 " , EdmSolution.Expand( " (2x-1)^7 " ));
62 Assert.AreEqual( " 81t^2 " , EdmSolution.Expand( " (9t-0)^2 " ));
63 }
64
65 [Test]
66 public void testANegative()
67 {
68 Assert.AreEqual( " 625m^4-1500m^3+1350m^2-540m+81 " , EdmSolution.Expand( " (-5m+3)^4 " ));
69 Assert.AreEqual( " -8k^3-36k^2-54k-27 " , EdmSolution.Expand( " (-2k-3)^3 " ));
70 Assert.AreEqual( " 1 " , EdmSolution.Expand( " (-7x-7)^0 " ));
71 Assert.AreEqual( " -35831808a^7+20901888a^6-5225472a^5+725760a^4-60480a^3+3024a^2-84a+1 " , EdmSolution.Expand( " (-12a+1)^7 " ));
72 Assert.AreEqual( " -184528125k^5-123018750k^4-32805000k^3-4374000k^2-291600k-7776 " , EdmSolution.Expand( " (-45k-6)^5 " ));
73 Assert.AreEqual( " -12c+1 " , EdmSolution.Expand( " (-12c+1)^1 " ));
74 Assert.AreEqual( " 100000000x^4+4000000x^3+60000x^2+400x+1 " , EdmSolution.Expand( " (-100x-1)^4 " ));
75 Assert.AreEqual( " -1000x^3+2400x^2-1920x+512 " , EdmSolution.Expand( " (-10x+8)^3 " ));
76 Assert.AreEqual( " -128w^7-448w^6-672w^5-560w^4-280w^3-84w^2-14w-1 " , EdmSolution.Expand( " (-2w-1)^7 " ));
77 Assert.AreEqual( " -n^5-60n^4-1440n^3-17280n^2-103680n-248832 " , EdmSolution.Expand( " (-n-12)^5 " )); // extra static test added by docgunthrop
78 Assert.AreEqual( " -k^7+28k^6-336k^5+2240k^4-8960k^3+21504k^2-28672k+16384 " , EdmSolution.Expand( " (-k+4)^7 " )); // extra static test added by docgunthrop
79 Assert.AreEqual( " 81t^2 " , EdmSolution.Expand( " (-9t-0)^2 " ));
80 }
81
82 private static readonly Random rand = new Random();
83 private static int rands( int limit)
84 {
85 return rand.Next( 2 * limit + 2 ) - limit;
86 }
87
88 private static string makeTestCase( int c, int n, int p)
89 {
90 int coeff = 0 ;
91 while (coeff == 0 )
92 coeff = rands(c);
93 return string .Format( " ({0}{1}{2:+0;-#})^{3} " , coeff == 1 ? "" : (coeff == - 1 ? " - " : "" + coeff), ( char )( ' a ' + rand.Next( 26 )), rands(n), rand.Next(p) + 2 );
94 }
95
96 [Test]
97 public void testRandom()
98 {
99
100 for ( int i = 0 ; i < 50 ; ++ i)
101 {
102 string eq = makeTestCase( 16 , 32 , 4 );
103 Assert.AreEqual(ReferenceSolution.Expand(eq), EdmSolution.Expand(eq), " Input: " + eq);
104 }
105
106 for ( int i = 0 ; i < 100 ; ++ i)
107 {
108 string eq = makeTestCase( 9 , 16 , 9 );
109 Assert.AreEqual(ReferenceSolution.Expand(eq), EdmSolution.Expand(eq), " Input: " + eq);
110 }
111 }
112
113 #region Reference solution
114 private class ReferenceSolution
115 {
116
117 private static readonly Regex re = new Regex( @" \((-?\d*)([a-z])([\+\-]\d+)\)\^(\d+) " );
118
119 public static string Expand( string expr)
120 {
121
122 Match m = re.Match(expr);
123
124 string sa = m.Groups[ 1 ].Value;
125 int a = ( "" .Equals(sa) ? 1 : ( " - " .Equals(sa) ? - 1 : int .Parse(sa)));
126
127 string x = m.Groups[ 2 ].Value;
128
129 string sb = m.Groups[ 3 ].Value;
130 int b = "" .Equals(sb) ? 0 : int .Parse(sb);
131
132 string se = m.Groups[ 4 ].Value;
133 int exp = "" .Equals(se) ? 1 : int .Parse(se);
134 if (exp == 0 )
135 return " 1 " ;
136
137 if (exp == 1 )
138 return sa + x + sb;
139
140 if (b == 0 )
141 {
142 long coeff = ( long )Math.Pow(a, exp);
143 return (coeff == 1 ? "" : (coeff == - 1 ? " - " : coeff.ToString())) + x + " ^ " + exp;
144 }
145
146 List< long > binoms = new List< long > ();
147 for ( int i = 0 ; i <= exp; ++ i)
148 binoms.Add(nk(exp, i));
149
150 long coeff1 = ( long )Math.Pow(a, exp);
151 StringBuilder terms = new StringBuilder();
152 for ( int i = exp; i >= 0 ; -- i)
153 {
154
155 long coeff = coeff1 * binoms[i];
156
157 if (i != exp && coeff > 0 )
158 terms.Append( ' + ' );
159
160 if (coeff < 0 )
161 terms.Append( ' - ' );
162
163 if ((coeff != 1 && coeff != - 1 ) || i == 0 )
164 terms.Append(coeff > 0 ? coeff : - coeff);
165
166 if (i > 0 )
167 terms.Append(x);
168
169 if (i > 1 )
170 terms.Append( " ^ " + i);
171
172 coeff1 = coeff1 / a * b;
173 }
174
175 return terms.ToString();
176 }
177
178 private static readonly List<List< long >> nka = new List<List< long >> ();
179
180 private static long nk( int n, int k)
181 {
182
183 if (n == 0 || k == 0 )
184 return 1 ;
185
186 if (k == 1 )
187 return n;
188
189 if (n - k < k)
190 return nk(n, n - k);
191
192 for ( int i = nka.Count; i <= n; ++ i)
193 nka.Add( new List< long > ());
194
195 List< long > ns = nka[n];
196 for ( int i = ns.Count; i <= k; ++ i)
197 ns.Add( 0L );
198
199 if (ns[k] != 0 )
200 return ns[k];
201 else
202 {
203 long b = nk(n - 1 , k - 1 ) + nk(n - 1 , k);
204 ns[k] = b;
205 return b;
206 }
207 }
208 }
209 #endregion
210 }
211 }
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