document.write( "Question 697241: Hello
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Algebra.Com's Answer #429901 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Using only the digits 0,1,2,3, write down all \r\n" );
document.write( "possible three digit numbers with sum of digits 3,\r\n" );
document.write( "allowing numbers to begin with 0, in order from \r\n" );
document.write( "largest to smallest:\r\n" );
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document.write( "300\r\n" );
document.write( "210\r\n" );
document.write( "201\r\n" );
document.write( "120\r\n" );
document.write( "111\r\n" );
document.write( "102\r\n" );
document.write( "030\r\n" );
document.write( "021\r\n" );
document.write( "012\r\n" );
document.write( "003\r\n" );
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document.write( "in each, use the digits of each 3-digit number \r\n" );
document.write( "in the order they appear as exponents of \r\n" );
document.write( "a, b, and c respectively, as this scheme shows:\r\n" );
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document.write( "300 --> a3b0c0 = a³\r\n" );
document.write( "210 --> a2b1c0 = a²b\r\n" );
document.write( "201 --> a2b0c1 = a²c\r\n" );
document.write( "120 --> a1b2c0 = ab²\r\n" );
document.write( "111 --> a1b1c1 = abc\r\n" );
document.write( "102 --> a1b0c2 = ac²\r\n" );
document.write( "030 --> a0b3c0 = b³\r\n" );
document.write( "021 --> a0b2c1 = b²c\r\n" );
document.write( "012 --> a0b1c2 = bc²\r\n" );
document.write( "003 --> a0b0c3 = c³\r\n" );
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document.write( "Rewrite the dividend a³+b³+c³-3abc putting in zero\r\n" );
document.write( "place-holders for each of those not represented in\r\n" );
document.write( "that dividend, in that order.  That is, the dividend \r\n" );
document.write( "becomes:\r\n" );
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document.write( "a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
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document.write( "So we start with this:\r\n" );
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document.write( "a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
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document.write( "Then we divide a³ by a, getting a², and we write that as the 1st term of the quotient.\r\n" );
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document.write( "          a²                                                            \r\n" );
document.write( "a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "         \r\n" );
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document.write( "Then we multiply that by each term of the divisor and place each product\r\n" );
document.write( "under the term that it is like, and draw a line under it:\r\n" );
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document.write( "          a²                                                            \r\n" );
document.write( "a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "          a³ +  a²b +  a²c\r\n" );
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document.write( "Then we subtract and bring EVERY term down\r\n" );
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document.write( "          a²                                                            \r\n" );
document.write( "a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "          a³ +  a²b +  a²c\r\n" );
document.write( "               -a²b -  a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
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document.write( "Then we divide -a²b by a, getting -ab, and we write that as the 2nd term of the quotient:\r\n" );
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document.write( "          a² - ab                                                       \r\n" );
document.write( "a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "          a³ +  a²b +  a²c\r\n" );
document.write( "               -a²b -  a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
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document.write( "Then we multiply that by each term of the divisor and place each product\r\n" );
document.write( "under the term that it is like, and draw a line under it, subtract and bring\r\n" );
document.write( "EVERY term down:\r\n" );
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document.write( "          a² - ab                                                       \r\n" );
document.write( "a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "          a³ +  a²b +  a²c\r\n" );
document.write( "               -a²b -  a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "               -a²b        -  ab² -  abc\r\n" );
document.write( "                      -a²c +  ab² - 2abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
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document.write( "Keep doing that, and end up with this:\r\n" );
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document.write( "          a² - ab - ac + b² - bc  + c²                                  \r\n" );
document.write( "a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "          a³ +  a²b +  a²c\r\n" );
document.write( "               -a²b -  a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "               -a²b        -  ab² -  abc\r\n" );
document.write( "                      -a²c +  ab² - 2abc + 0ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "                      -a²c        -  abc -  ac²\r\n" );
document.write( "                              ab² -  abc +  ac² +  b³ + 0b²c + 0bc² + c³\r\n" );
document.write( "                              ab²               +  b³ +  b²c\r\n" );
document.write( "                                  -  abc +  ac² - 0b³ -  b²c + 0bc² + c³\r\n" );
document.write( "                                  -  abc              -  b²c -  bc²\r\n" );
document.write( "                                            ac² - 0b³ + 0b²c +  bc² + c³\r\n" );
document.write( "                                            ac²              +  bc² + c³\r\n" );
document.write( "                                                                       0\r\n" );
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document.write( "So the answer is: a² - ab - ac + b² - bc  + c²\r\n" );
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document.write( "Edwin
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