document.write( "Question 458892: Given the polynomial function f(x), find the rational zeros, then the other zeros(that is, solve the equation f(x)=0), and factor f(x) into linear factors.)\r
\n" ); document.write( "\n" ); document.write( "f(x)=3x^4-10x^3+20x^2-40x+32\r
\n" ); document.write( "\n" ); document.write( "I know that the first zero is 2 and that their should be 4 zeros in all.\r
\n" ); document.write( "\n" ); document.write( "thanks for the help. \n" ); document.write( "
Algebra.Com's Answer #314776 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The other tutor did not factor it completely into\r\n" );
document.write( "linear factors.\r\n" );
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document.write( "f(x)=3x^4-10x^3+20x^2-40x+32\r\n" );
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document.write( "You say you know that the first zero is 2,\r\n" );
document.write( "so we do this synthetic division:\r\n" );
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document.write( "2|3 -10  20 -40  32\r\n" );
document.write( " |    6  -8  24 -32\r\n" );
document.write( "  3  -4  12 -16   0\r\n" );
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document.write( "So we have now partially factored f(x) as\r\n" );
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document.write( "f(x) = (x-2)(3x³-4x²+12x-16)\r\n" );
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document.write( "Now we will factor this\r\n" );
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document.write( "3x³-4x²+12x-16\r\n" );
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document.write( "by grouping.  Out of the first two terms \r\n" );
document.write( "factor out x²\r\n" );
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document.write( "x²(3x-4)+12x-16\r\n" );
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document.write( "Out of the last two terms factor out +4\r\n" );
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document.write( "x²(3x-4)+4(3x-4)\r\n" );
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document.write( "Factor out (3x-4) and get\r\n" );
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document.write( "(3x-4)(x²+4)\r\n" );
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document.write( "So we have now further partially factored f(x) as\r\n" );
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document.write( "f(x) = (x-2)(3x-4)(x²+4)\r\n" );
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document.write( "Now since i² = -1, -i² = 1, and since 4 = 4*1, \r\n" );
document.write( "4*1 also equals 4(-i²) or -4i²,\r\n" );
document.write( "so replace 4 by -4i²\r\n" );
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document.write( "f(x) = (x-2)(3x-4)(x²-4i²)\r\n" );
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document.write( "Now we can completely factor f(x) by factoring that\r\n" );
document.write( "last factor as the difference of two squares:\r\n" );
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document.write( "f(x) = (x-2)(3x-4)(x-2i)(x+2i)\r\n" );
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document.write( "To solve it we set each of the factors = 0\r\n" );
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document.write( "x-2=0, 3x-4=0,   x-2i=0,  x+2i=0\r\n" );
document.write( "  x=2,   3x=4,      x=2i     x=-2i\r\n" );
document.write( "          x=4/3\r\n" );
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document.write( "So the solutions are 2, 4/3, 2i, and -2i.\r\n" );
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document.write( "Edwin
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