document.write( "Question 36358: 1.If α, b, g are the roots of the equation
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\n" ); document.write( " x^3─7x^2 + x + 5 = 0
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\n" ); document.write( " Find the equation whose roots are
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\n" ); document.write( " α^2 + b^2, b^2 + g^2, g^2 + α^2.\r
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\n" ); document.write( "\n" ); document.write( "2.Find all the fifth roots of (2+i).
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Algebra.Com's Answer #22376 by khwang(438)\"\" \"About 
You can put this solution on YOUR website!
1.If α, b, g are the roots of the equation\r
\n" ); document.write( "\n" ); document.write( "x^3─7x^2 + x + 5 = 0\r
\n" ); document.write( "\n" ); document.write( "Find the equation whose roots are\r
\n" ); document.write( "\n" ); document.write( "α^2 + b^2, b^2 + g^2, g^2 + α^2. \r
\n" ); document.write( "\n" ); document.write( "Sol: since α + b+ g= 7,
\n" ); document.write( " αb + bg+ gα = 1 and
\n" ); document.write( " αbg = - 5.
\n" ); document.write( " use α^2+ b^2 + g^2 = (α+b+g)^2 - 2(αb + bg+ gα) = 49 - 2 = 47.
\n" ); document.write( " (α^2 + b^2)(b^2 + g^2)+ (b^2 + g^2)(g^2 + α^2) + (g^2 + α^2)(α^2 + b^2)
\n" ); document.write( " = (47- g^2)(47- α^2) + (47- α^2)(47- b^2) + (47- b^2)(47- g^2)
\n" ); document.write( " = 3*47^2 - 47*2(α^2+ b^2 + g^2) + α^2 g^2 + α^2 b^2 + b^2 g^2
\n" ); document.write( " where α^2 g^2 + α^2 b^2 + b^2 g^2 = (αb + bg+ gα)^2 - 2αbg(α + b+ g)
\n" ); document.write( " = 1 + 10*7 = 71.
\n" ); document.write( " So, (α^2 + b^2)(b^2 + g^2)+ (b^2 + g^2)(g^2 + α^2) + (g^2 + α^2)(α^2 + b^2)
\n" ); document.write( " = 3*47^2 - 2*47^2 + 71 = 2280
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\n" ); document.write( " And, (α^2 + b^2)(b^2 + g^2)(g^2+ α^2)
\n" ); document.write( " = (47- α^2)(47- g^2) (47- b^2)
\n" ); document.write( "= 47^3 - 47(α^2+ b^2 + g^2) + 47(α^2 b^2 + g^2b^2+ α^2g^2)- α^2 b^2 g^2
\n" ); document.write( "= 47^3 - 47^2 + 47*71- 25 = 104926\r
\n" ); document.write( "\n" ); document.write( " Hence, the required equation as:
\n" ); document.write( " x^3 - 47x^2 + 2280 x - 104926 = 0.\r
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\n" ); document.write( "\n" ); document.write( "2.Find all the fifth roots of (2+i).\r
\n" ); document.write( "\n" ); document.write( " Use de Moevie(??) law;
\n" ); document.write( " 2+i = r (cos t+ i sin t), where r = sqrt(2^2+1) = sqrt(5)
\n" ); document.write( " and t = ArcTan 1/2
\n" ); document.write( " the five 5th roots are r^(1/5) (cos \"%282pi+k+%2B+t%29%2F+5%29\" +
\n" ); document.write( " i sin \"%282pi+k+%2B+t%29%2F+5+\" }
\n" ); document.write( " for k = 0,1,2,3,4.
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\n" ); document.write( " or 5^(1/10) (cos \"t%2F+5%29\" + i sin \"t%2F+5+\" },
\n" ); document.write( " 5^(1/10) (cos \"%282pi++%2B+t%29%2F+5%29\" + i sin \"%282pi++%2B+t%29%2F+5+\" },
\n" ); document.write( " 5^(1/10) (cos \"%284pi++%2B+t%29%2F+5%29\" + i sin \"%284pi++%2B+t%29%2F+5+\" },
\n" ); document.write( " 5^(1/10) (cos \"%286pi++%2B+t%29%2F+5%29\" + i sin \"%286pi++%2B+t%29%2F+5+\" } ,
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\n" ); document.write( " 5^(1/10) (cos \"%288pi++%2B+t%29%2F+5%29\" + i sin \"%288pi++%2B+t%29%2F+5+\" }\r
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