document.write( "Question 1140166: 1. factorise x^4-5x^3+5x^2+5x-6
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Algebra.Com's Answer #760665 by Edwin McCravy(20056)\"\" \"About 
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1.   Factor:  \"x%5E4-5x%5E3%2B5x%5E2%2B5x-6\"\r\n" );
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document.write( "All potential roots or zeros are ± the factors of 6, which are ±1,±2,±3,±6\r\n" );
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document.write( "We try the easiest one first, which is 1 with synthetic division,  We are\r\n" );
document.write( "dividing by x-1\r\n" );
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document.write( "1 | 1 -5  5  5 -6\r\n" );
document.write( "  |    1 -4  1  6\r\n" );
document.write( "    1 -4  1  6  0\r\n" );
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document.write( "Luckily that gave 0 remainder, telling us that 1 is a zero or root and that\r\n" );
document.write( "x-1 is a factor,  So the first partial factorization using (x-1) and the bottom\r\n" );
document.write( "numbers is:\r\n" );
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document.write( "(x-1)(x³-4x²+x+6)\r\n" );
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document.write( "Next we factor x³-4x²+x+6\r\n" );
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document.write( "All potential roots or zeros are again ± the factors of 6, which are ±1,±2,±3,±6\r\n" );
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document.write( "We try the easiest one first again, which is 1, with synthetic division,  We are\r\n" );
document.write( "again dividing by x-1\r\n" );
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document.write( "1 | 1 -4  1  6\r\n" );
document.write( "  |    1 -3 -2\r\n" );
document.write( "    1 -3 -2  4\r\n" );
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document.write( "That does not give 0 remainder, so we try the next easiest one, which is -1,\r\n" );
document.write( "with synthetic division,  We are dividing this time by x+1\r\n" );
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document.write( "-1 | 1 -4  1  6\r\n" );
document.write( "   |   -1  5 -6\r\n" );
document.write( "     1 -5  6  0\r\n" );
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document.write( "Luckily that gave 0 remainder, telling us that -1 is a zero or root and that\r\n" );
document.write( "x+1 is a factor,  So the next partial factorization using (x+1) and the bottom\r\n" );
document.write( "numbers is:\r\n" );
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document.write( "(x-1)(x+1)(x²-5x+6)\r\n" );
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document.write( "Now we can complete the factorization without synthetic division:\r\n" );
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document.write( "(x-1)(x+1)(x-2)(x-3)\r\n" );
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document.write( "Factor:  \"x%5E9-y%5E9\", we rewrite as \"%28x%5E3%29%5E3%2B%28y%5E3%29%5E3\" and\r\n" );
document.write( "then we can use the formula: \"A%5E3-B%5E3=%28A-B%29%28A%5E2%2BAB%2BB%5E2%29\"\r\n" );
document.write( "with A=x³ and B=y³\r\n" );
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document.write( "\"A%5E3-B%5E3=%28A-B%29%28A%5E2%2BAB%2BB%5E2%29\"\r\n" );
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document.write( "\"x%5E9-y%5E9=%28x%5E3-y%5E3%29%28x%5E6%2Bx%5E3y%5E3%2By%5E6%29\"\r\n" );
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document.write( "3.solve the equation \"x%5E4-5x%5E3%2B5x%5E2%2B5x-6=0\"\r\n" );
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document.write( "We have already factored the left side in the first problem, so\r\n" );
document.write( "we have:\r\n" );
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document.write( "(x-1)(x+1)(x-2)(x-3) = 0\r\n" );
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document.write( "We set each factor equal to 0:\r\n" );
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document.write( "x-1=0;  x+1=0;  x-2=0;  x-3=0\r\n" );
document.write( "  x=1;    x=-1;   x=2;    x=3\r\n" );
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document.write( "Those are the four solutions.\r\n" );
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document.write( "4.  \"7e%5Ex+-+3+=+2e%5E%282x%29\"\r\n" );
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document.write( "Let ex = u, then e2x = u2\r\n" );
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document.write( "    \"7u-3=2u%5E2\"\r\n" );
document.write( "    \"0=2u%5E2-7u%2B3\"\r\n" );
document.write( "    \"2u%5E2-7u%2B3=0\"\r\n" );
document.write( "    \"%282u-1%29%28u-3%29=0\"\r\n" );
document.write( "    2u-1=0;   u-3=0\r\n" );
document.write( "      2u=1      u=3\r\n" );
document.write( "       u=\"1%2F2\"\r\n" );
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document.write( "Substitute ex for u\r\n" );
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document.write( "\"e%5Ex=1%2F2\";   \"e%5Ex=3\"\r\n" );
document.write( "\"x=ln%281%2F2%29\";  \"x=ln%283%29\"\r\n" );
document.write( "\"x=ln%282%5E%28-1%29%29\";\r\n" );
document.write( "\"x=-ln%282%29\"\r\n" );
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document.write( "Here ln() means natural logarithm.\r\n" );
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document.write( "Edwin
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