document.write( "Question 155001: Directions state: Factor the polynomials.\r
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Algebra.Com's Answer #114119 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Factor \r\n" );
document.write( "\"x%5E3+%2B+27\"\r\n" );
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document.write( "Let's write it this way\r\n" );
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document.write( "\"x%5E3+%2B+3%5E3\"\r\n" );
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document.write( "That is the sum of two cubes.\r\n" );
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document.write( "Now what if we dropped those cubes and had \"x%2B3\"\r\n" );
document.write( "Then suppose we divided that into \"x%5E3+%2B+3%5E3\" by \r\n" );
document.write( "long division:\r\n" );
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document.write( "            x² - 3x +  9\r\n" );
document.write( "     ------------------- \r\n" );
document.write( "x + 3)x³ + 0x² + 0x + 27\r\n" );
document.write( "      x² + 3x²\r\n" );
document.write( "      --------\r\n" );
document.write( "          -3x² + 0x\r\n" );
document.write( "          -3x² - 9x\r\n" );
document.write( "          ---------\r\n" );
document.write( "                 9x + 27\r\n" );
document.write( "                 9x + 27  \r\n" );
document.write( "                 -------\r\n" );
document.write( "                       0 \r\n" );
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document.write( "That gives a zero remainder. So you now \r\n" );
document.write( "know from that, that\r\n" );
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document.write( "x³ + 27\r\n" );
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document.write( "or\r\n" );
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document.write( "x³ + 3³\r\n" );
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document.write( "factors as \r\n" );
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document.write( "(x + 3)(x² - 3x + 9)\r\n" );
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document.write( "But if you learn the principle then you\r\n" );
document.write( "wouldn't have to use long division. Sure, if\r\n" );
document.write( "you forgot the principle you could use\r\n" );
document.write( "long division every time.  But you should\r\n" );
document.write( "memorize the principle to save time.\r\n" );
document.write( "\r\n" );
document.write( "The principle is\r\n" );
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document.write( "When you have the sum of two cubes\r\n" );
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document.write( "\"FIRST%5E3+%2B+SECOND%5E3\"\r\n" );
document.write( "\r\n" );
document.write( "it factors as \r\n" );
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document.write( "\"%28FIRST%2BSECOND%29%28FIRST%5E2-FIRST%2ASECOND%2BSECOND%5E2%29\" \r\n" );
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document.write( "So in the case of\r\n" );
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document.write( "\"x%5E3+%2B+27\"\r\n" );
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document.write( "You write it as\r\n" );
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document.write( "\"x%5E3+%2B+3%5E3\"\r\n" );
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document.write( "Then \"FIRST+=+x\" and \"SECOND=3\"\r\n" );
document.write( "\r\n" );
document.write( "so\r\n" );
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document.write( "\"%28FIRST%2BSECOND%29%28FIRST%5E2-FIRST%2ASECOND%2BSECOND%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "becomes\r\n" );
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document.write( "\"%28x%2B3%29%28x%5E2-x%2A3%2B3%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%2B3%29%28x%5E2-3x%2B9%29\"\r\n" );
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document.write( "Then you don't have to use long division.\r\n" );
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document.write( "-----------------------------------------------\r\n" );
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document.write( "Suppose, instead it were\r\n" );
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document.write( "\"x%5E3+-+27\"\r\n" );
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document.write( "Let's write it this way\r\n" );
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document.write( "\"x%5E3+-+3%5E3\"\r\n" );
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document.write( "That is the DIFFERENCE of two cubes.\r\n" );
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document.write( "Now what if we dropped those cubes and had \"x-3\"\r\n" );
document.write( "Then suppose we divided that into \"x%5E3+-+3%5E3\" by \r\n" );
document.write( "long division:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "            x² + 3x +  9\r\n" );
document.write( "     ------------------- \r\n" );
document.write( "x - 3)x³ + 0x² + 0x + 27\r\n" );
document.write( "      x² - 3x²\r\n" );
document.write( "      --------\r\n" );
document.write( "           3x² + 0x\r\n" );
document.write( "           3x² - 9x\r\n" );
document.write( "          ---------\r\n" );
document.write( "                 9x + 27\r\n" );
document.write( "                 9x + 27  \r\n" );
document.write( "                 -------\r\n" );
document.write( "                       0 \r\n" );
document.write( "\r\n" );
document.write( "That gives a zero remainder. So you now \r\n" );
document.write( "know from that, that\r\n" );
document.write( "\r\n" );
document.write( "x³ - 27\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "x³ - 3³\r\n" );
document.write( "\r\n" );
document.write( "factors as \r\n" );
document.write( "\r\n" );
document.write( "(x - 3)(x² + 3x + 9)\r\n" );
document.write( "\r\n" );
document.write( "But if you learn the principle then you\r\n" );
document.write( "wouldn't have to use long division. Sure, if\r\n" );
document.write( "you forgot the principle you could use\r\n" );
document.write( "long division every time.  But you should\r\n" );
document.write( "memorize the principle to save time.\r\n" );
document.write( "\r\n" );
document.write( "The principle is\r\n" );
document.write( "\r\n" );
document.write( "When you have the sum of two cubes\r\n" );
document.write( "\r\n" );
document.write( "\"FIRST%5E3+-+SECOND%5E3\"\r\n" );
document.write( "\r\n" );
document.write( "it factors as \r\n" );
document.write( "\r\n" );
document.write( "\"%28FIRST-SECOND%29%28FIRST%5E2%2BFIRST%2ASECOND%2BSECOND%5E2%29\" \r\n" );
document.write( "\r\n" );
document.write( "So in the case of\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E3+-+27\"\r\n" );
document.write( "\r\n" );
document.write( "You write it as\r\n" );
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document.write( "\"x%5E3+-+3%5E3\"\r\n" );
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document.write( "Then \"FIRST+=+x\" and \"SECOND=3\"\r\n" );
document.write( "\r\n" );
document.write( "so\r\n" );
document.write( "\r\n" );
document.write( "\"%28FIRST-SECOND%29%28FIRST%5E2%2BFIRST%2ASECOND%2BSECOND%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "becomes\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-3%29%28x%5E2%2Bx%2A3%2B3%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-3%29%28x%5E2%2B3x%2B9%29\"\r\n" );
document.write( "\r\n" );
document.write( "Then you don't have to use long division.\r\n" );
document.write( "\r\n" );
document.write( "---------\r\n" );
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document.write( "In general\r\n" );
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document.write( "\"FIRST%5E3+%2B-+SECOND%5E3\"\r\n" );
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document.write( "factors as\r\n" );
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document.write( "\"%28FIRST%2B-SECOND%29%28FIRST%5E2%2B-FIRST%2ASECOND%2BSECOND%5E2%29\"\r\n" );
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document.write( "Edwin
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