document.write( "Question 223550: x+1=9x^3 + 9x^2 solve the polynomial equation.\r
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Algebra.Com's Answer #167151 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
x+1=9x^3 + 9x^2 solve the polynomial equation.
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document.write( "No, it only needs 3 \"answers\", \"solutions\" or \"roots\", whichever\r\n" );
document.write( "label you wish to give them.  That's because the degree (largest\r\n" );
document.write( "exponent of x) is 3.\r\n" );
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document.write( "\"x%2B1=9x%5E3+%2B+9x%5E2\"\r\n" );
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document.write( "Get zero on the right:\r\n" );
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document.write( "\"-9x%5E3+-+9x%5E2%2Bx%2B1=0\"\r\n" );
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document.write( "Multiply through by -1 so leading coefficient will be\r\n" );
document.write( "positive:\r\n" );
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document.write( "\"9x%5E3+%2B+9x%5E2-x-1=0\"\r\n" );
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document.write( "Possible rational zeros are\r\n" );
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document.write( "±1, ±\"1%2F3\", and ±\"1%2F9\"\r\n" );
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document.write( "By DesCartes' rule of signs, there is one positive\r\n" );
document.write( "answer (solution or root), and 2 or 0 negative ones.\r\n" );
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document.write( "Let's try x=1, that is, divide by x-1 to see if\r\n" );
document.write( "1 is a solution:\r\n" );
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document.write( "1 | 9   9   -1   -1\r\n" );
document.write( "  |     9  -18  -19\r\n" );
document.write( "   ----------------\r\n" );
document.write( "    9 -18  -19  -20\r\n" );
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document.write( "Nope, the remainder is -20, not zero, so x-1 is not a\r\n" );
document.write( "divisor of \"9x%5E3+%2B+9x%5E2-x-1\", and so x=1 is not\r\n" );
document.write( "a solution.\r\n" );
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document.write( "Let's try x=-1, that is, divide by x+1 to see if\r\n" );
document.write( "-1 is a solution:\r\n" );
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document.write( "-1 | 9   9  -1 -1\r\n" );
document.write( "   |    -9   0  1\r\n" );
document.write( "   --------------\r\n" );
document.write( "     9   0  -1  0\r\n" );
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document.write( "Aha! The remainder is zero, so x+1 is a divisor of\r\n" );
document.write( "of \"9x%5E3+%2B+9x%5E2-x-1\", and therefore x=-1 is a\r\n" );
document.write( "solution.\r\n" );
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document.write( "Anytime we divide by synthetic division and get a zero \r\n" );
document.write( "remainder, we have factored the polynomial.  In this\r\n" );
document.write( "case we have factored \"9x%5E3+%2B+9x%5E2-x-1\" and now\r\n" );
document.write( "we have\r\n" );
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document.write( "\"%28x%2B1%29%289x%5E2%2B0x-1%29=0\"\r\n" );
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document.write( "or eliminating the \"0x\" term:\r\n" );
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document.write( "\"%28x%2B1%29%289x%5E2-1%29=0\"\r\n" );
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document.write( "Now we can factor the expression in the second parentheses\r\n" );
document.write( "as the difference of two perfect squares and get:\r\n" );
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document.write( "\"%28x%2B1%29%283x-1%29%283x%2B1%29=0\"\r\n" );
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document.write( "Setting each factor = 0,\r\n" );
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document.write( "x+1=0 gives x=-1\r\n" );
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document.write( "3x-1=0 gives x=\"1%2F3\"\r\n" );
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document.write( "3x+1=0 gives x=\"-1%2F3\"\r\n" );
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document.write( "Edwin
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