document.write( "Question 550815: Find all the rational zeros of the function f(x)=8x^4-24x^3-424x^2-72x
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Algebra.Com's Answer #359129 by Edwin McCravy(20056)\"\" \"About 
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document.write( "f(x) = 8x4 - 24x3 - 424x2 - 72x\r\n" );
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document.write( "We set the right side = 0\r\n" );
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document.write( "8x4 - 24x3 - 424x2 - 72x = 0\r\n" );
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document.write( "We can factor out 8x:\r\n" );
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document.write( "8x(x3 - 3x2 - 53x - 9) = 0\r\n" );
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document.write( "So by the zero factor principle:\r\n" );
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document.write( "8x = 0         x3 - 3x2 - 53x - 9 = 0\r\n" );
document.write( " x = 0\r\n" );
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document.write( "So we see that one rational solution is 0\r\n" );
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document.write( "Next we try to solve x3 - 3x2 - 53x - 9 = 0\r\n" );
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document.write( "If it has any rational solutions they must be\r\n" );
document.write( "± some divisor of 9\r\n" );
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document.write( "Possible rational solutions of x3 - 3x2 - 53x - 9 = 0\r\n" );
document.write( "are ±1, ±3, ±9\r\n" );
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document.write( "It has 1 sign change so we know it has one positive\r\n" );
document.write( "solution.\r\n" );
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document.write( "Try x = 1\r\n" );
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document.write( "1 |1 -3 -53  -9\r\n" );
document.write( "  |   1  -2 -55\r\n" );
document.write( "   1 -2 -55 -64\r\n" );
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document.write( "No, 1 is not a solution, since the remainder is not 0\r\n" );
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document.write( "Try x = 3\r\n" );
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document.write( "3 |1 -3 -53   -9\r\n" );
document.write( "  |   3   0 -159\r\n" );
document.write( "   1  0 -53 -168\r\n" );
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document.write( "No, 3 is not a solution, since the remainder is not 0\r\n" );
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document.write( "Try x = 9\r\n" );
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document.write( "9 |1 -3 -53 -9\r\n" );
document.write( "  |   9  54  9\r\n" );
document.write( "   1  6   1  0\r\n" );
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document.write( "Hurray! 9 is a solution because the remainder is 0.\r\n" );
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document.write( "Now we know that 0 and 3 are rational zeros of f(x).\r\n" );
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document.write( "Now we have factored f(x) as\r\n" );
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document.write( "f(x) = 8x(x - 9)(x2 + 6x + 1)\r\n" );
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document.write( "So all the rational zeros of f(x) are 0 and 9\r\n" );
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document.write( "However there are two irrational solutions obtainable by\r\n" );
document.write( "setting x2 + 6x + 1 = 0 and using the quadratic formula:\r\n" );
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document.write( "x = \"%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\" \r\n" );
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document.write( "x = \"%28-6+%2B-+sqrt%28+6%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29+\" \r\n" );
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document.write( "x = \"%28-6+%2B-+sqrt%28+36-4+%29%29%2F2+\"\r\n" );
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document.write( "x = \"%28-6+%2B-+sqrt%28+32+%29%29%2F2+\"\r\n" );
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document.write( "x = \"%28-6+%2B-+sqrt%2816%2A2+%29%29%2F2+\"\r\n" );
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document.write( "x = \"%28-6+%2B-+4sqrt%282%29%29%2F2+\"\r\n" );
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document.write( "x =  \"%282%28-3+%2B-+2sqrt%282%29%29%29%2F2+\"\r\n" );
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document.write( "x =  \"%28cross%282%29%28-3+%2B-+2sqrt%282%29%29%29%2Fcross%282%29+\"\r\n" );
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document.write( "x =  \"-3+%2B-+2sqrt%282%29\"\r\n" );
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document.write( "But you weren't asked for those two irrational zeros.\r\n" );
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document.write( "The only rational zeros are 0 and 9.\r\n" );
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document.write( "Edwin
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