roblem 3 Find the solutions to the general cubic a x^3 +b x^2+c x +d=0 and...

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roblem 3 Find the solutions to the general cubic a x^3 +b x^2+cx +d=0 and the solutions to the general quartic a x^4+b x^3+c x^2+dx+e=0. Remember to put a space between your letters. The solutionsto the general quartic goes on for two pages it is a good idea tomaximize your page to see it. It is a theorem in modern abstractalgebra that there is no solution to the general quintic in termsof radicals.
Please write it clearly! thank you!

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Solution for Cubic Equation Lets say you have the equation ax3 bx2 cx d 0 The first thing to do is to get rid of the a out in front by dividing the whole equation by it Then we get something in the form of x3 ex2 fx g 0 The next thing we do is to get rid of the x2 term by replacing x with y e3 The new equation y3 py q 0 has no y2 term This is much easier to solve although its still kind of hard and it took a lot of really smart people a long time to do Lets introduce two new variables t and u defined by the equations u t q and tu p33 If youre really interested you might want to know what t and u actually are in terms of p and q 3qSqrt3 Sqrt4p3 27 q2 t 6 Sqrt3 3qSqrt3 Sqrt4p3 27q2 u 6 Sqrt3 That is either or for BOTH of the equations at the same time You cant have one as a and the other as a Anyway y1 CubeRoott CubeRootu will be a solution of y3 py q 0 Verify this result now DONT use the explicit formulas for t and u just plug CubeRoott CubeRootu into the cubic polynomial and make sure you see why it works To find the other two solutions of See Answer

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