AbramowitzStegunRoots[1] - conjugate roots t r2=O all roots...

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j '. " , " From Milton Apramowitz: 8i Irene Stegl:: Handbook of Mathematical Functions ' National Bureau of Standards (now National Institute of StanØards and Technology) 1964 . . . '~ 3.8. Algebraic Equations Solution of Qu~dratic Equations 3.8.1 Given az2+bz+c=O, Zi 2=-(.!):fl. ql n=b2-4ac . 2a 2a n. , Solution of Quartic Equations Z¡+z2=-b/a, Z¡z2=cja If 2)-0, two real roots, 2=0, two equal roots, (l-(O, pair of complex conjugate roots. 3.8.3 Givenz'+aaz3+aiz2+a¡z+aa=O, fid the real root Ui of the cubic equation ua-aiu2+ (aia:-4ao)u- (a~+ao-4aol)=O and determie the foll roots of the quartic as solutions of the two quadratic equations raa (~ )lJ u r(U)2 Jl v2+L"2=F 4+Ui-~ v+ 2i=FL 2¡ -aa =0 Solution of Cubic Equations i " 3.8.2 Given z3+aiz2+a¡z+ao=O, let 1 1 2 1 1 q=3 ai-9.~j r=6 (a¡ai-3ao)-27 ~. If t+r2)-O, one real root and a pair of complex
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Unformatted text preview: conjugate roots, t+r2=O, all roots real and at least two are equal, t+r2-(O, all roots real (ireducible case).- If all roots of the cubic equation are real use *the valu.e of 'U¡ ,:hieh gives real coeffcicnts ii~ tIll quadratlc cqUl1tlOn l1nd select signs so thl1t if z'+aaz3+ai¿+a¡z+ao= (z2+p¡z+q¡) (z2+P2Z+q2), then p¡ +p2=aa,p¡P2+q¡ +q2=ai,p¡q2+p2q¡=a¡, q¡q2=aa. If Z¡, .Z2, Za, r Z4 are the roots, 8i=(r+ (t+r2)lJl, 82=(r- (t+r2)l)l i:Zt= -a:, i:ztZjz/:= -a¡, i:ztZj=ai, Z¡Z2ZaZ4=/L. Let then ,;,' Z¡ = (8¡ +82) _ a: . . 3 1 ( a:i. . Z2=-2 8¡+82)-3+2 (8¡-82) 1 ai i., Za=-2 (81+82)-3-2 (8i-1I2)' If Zh Z2, za. are the roots of the cubic equation Zi+Z2+Za=-ai Z1Z2+ZiZa+Z2Za=a¡ Z1Z2Za= -aa...
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This note was uploaded on 11/02/2009 for the course APMA 2102 taught by Professor Keyes during the Spring '08 term at Columbia.

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