# SOLUTION: 1.If &#945;, b, g are the roots of the equation x^3&#9472;7x^2 + x + 5 = 0 Find the equation whose roots are &#945;^2 + b^2, b^2 + g^2, g^2 + &#945;^2. 2.Find

Algebra ->  Algebra  -> Square-cubic-other-roots -> SOLUTION: 1.If &#945;, b, g are the roots of the equation x^3&#9472;7x^2 + x + 5 = 0 Find the equation whose roots are &#945;^2 + b^2, b^2 + g^2, g^2 + &#945;^2. 2.Find      Log On

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 Click here to see ALL problems on Square-cubic-other-roots Question 36358: 1.If α, b, g are the roots of the equation x^3─7x^2 + x + 5 = 0 Find the equation whose roots are α^2 + b^2, b^2 + g^2, g^2 + α^2. 2.Find all the fifth roots of (2+i). Answer by khwang(438)   (Show Source): You can put this solution on YOUR website!1.If α, b, g are the roots of the equation x^3─7x^2 + x + 5 = 0 Find the equation whose roots are α^2 + b^2, b^2 + g^2, g^2 + α^2. Sol: since α + b+ g= 7, αb + bg+ gα = 1 and αbg = - 5. use α^2+ b^2 + g^2 = (α+b+g)^2 - 2(αb + bg+ gα) = 49 - 2 = 47. (α^2 + b^2)(b^2 + g^2)+ (b^2 + g^2)(g^2 + α^2) + (g^2 + α^2)(α^2 + b^2) = (47- g^2)(47- α^2) + (47- α^2)(47- b^2) + (47- b^2)(47- g^2) = 3*47^2 - 47*2(α^2+ b^2 + g^2) + α^2 g^2 + α^2 b^2 + b^2 g^2 where α^2 g^2 + α^2 b^2 + b^2 g^2 = (αb + bg+ gα)^2 - 2αbg(α + b+ g) = 1 + 10*7 = 71. 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) = 3*47^2 - 2*47^2 + 71 = 2280 And, (α^2 + b^2)(b^2 + g^2)(g^2+ α^2) = (47- α^2)(47- g^2) (47- b^2) = 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 = 47^3 - 47^2 + 47*71- 25 = 104926 Hence, the required equation as: x^3 - 47x^2 + 2280 x - 104926 = 0. 2.Find all the fifth roots of (2+i). Use de Moevie(??) law; 2+i = r (cos t+ i sin t), where r = sqrt(2^2+1) = sqrt(5) and t = ArcTan 1/2 the five 5th roots are r^(1/5) (cos + i sin } for k = 0,1,2,3,4. or 5^(1/10) (cos + i sin }, 5^(1/10) (cos + i sin }, 5^(1/10) (cos + i sin }, 5^(1/10) (cos + i sin } , and 5^(1/10) (cos + i sin } Kenny