Question

How to find zeris of bioquadratic equatiin?

Answer 1

Solving a quartic equationSpecial casesConsider the quarticIfthen,so zero is a root. To find the other roots, we can divide by  and solve the resultin cubic equationEvident roots: 1 and −1 and −kIf         then       ,    so 1 is a root. Similarly, ifthat is,then -1 is a root.When 1 is a root, we can divide by and getwhere  is a cubic polynomial, which may be solved to find  's other roots. Similarly, if -1 is a root,where  is some cubic polynomial.Ifthen −k is a root and we can factor out ,And ifthen both  and  are roots Now we can factor out  and getTo get Q 's other roots, we simply solve the quadratic factor.Biquadratic equationsIf  thenWe call such a polynomial a biquadratic, which is easy to solve.Let  Then Q becomes a quadratic q in z,Let  and  be the roots of q. Then the roots of our quartic Q areQuasi-symmetric equationsSteps:1) Divide by x 2.2) Use variable change z = x + m/x.The general case, along Ferrari's linesTo begin, the quartic must first be converted to a depressed quartic.Converting to a depressed quarticLetbe the general quartic equation which it is desired to solve. Divide both sides by A,The first step should be to eliminate the x3 term. To do this, change variables from x to u, such that.ThenExpanding the powers of the binomials producesCollecting the same powers of u yieldsNow rename the coefficients of u. LetThe resulting equation iswhich is a depressed quartic equation.If  then we have a biquadratic Equation, which (as explained above) is easily solved; using reverse substitution we can find our values for x.If  then one of the roots is  and the other roots can be found by dividing by u, and solving the resulting equation,