Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 2757. The discriminant of the cubic equation isShow that y 3 + py + q = 0∆ = p327 + q24 .(a) has three real roots, at least two of which are equal, if ∆ = 0.(b) has one real root and two conjugate imaginary roots if ∆ > 0.(c) has three distinct real roots if ∆ < 0.8. Solve the following cubic equations.(a) x 3 − 4x 2 + 11x + 30 = 0(b) x 3 − 3x + 5 = 0(c) x 3 − 3x + 2 = 0(d) x 3 + x + 3 = 09. Show that the general quartic equationcan be reduced tox 4 + ax 3 + bx 2 + cx + d = 0y 4 + py 2 + qy + r = 0by using the substitution x = y − a/4.10. Show that (y 2 + 1 ) 2 ( ) 12 z = (z − p)y 2 − qy +4 z2 − r .11. Show that the right-hand side of (10) can be put in the form (my + k) 2 ifand only if( ) 1q 2 − 4(z − p)4 z2 − r = 0.12. From (11) obtain the resolvent cubic equationz 3 − pz 2 − 4rz + (4pr − q 2 ) = 0.Solving the resolvent cubic equation, put the equation found in (10) in theform (y 2 + 1 2 z ) 2= (my + k) 2to obtain the solution of the quartic equation.13. Use this method to solve the following quartic equations.