Abstract Algebra Theory and Applications - Computer Science ...

274 CHAPTER 15 POLYNOMIALSAdditional Exercises: Solving the Cubic and QuarticEquations1. Solve the general quadratic equationto obtainax 2 + bx + c = 0x = −b ± √ b 2 − 4ac.2aThe discriminant of the quadratic equation ∆ = b 2 − 4ac determines thenature of the solutions of the equation. If ∆ > 0, the equation has twodistinct real solutions. If ∆ = 0, the equation has a single repeated real root.If ∆ < 0, there are two distinct imaginary solutions.2. Show that any cubic equation of the formx 3 + bx 2 + cx + d = 0can be reduced to the form y 3 + py + q = 0 by making the substitutionx = y − b/3.3. Prove that the cube roots of 1 are given by4. Make the substitutionω = −1 + i√ 32ω 2 = −1 − i√ 32ω 3 = 1.y = z − p3zfor y in the equation y 3 +py +q = 0 and obtain two solutions A and B for z 3 .5. Show that the product of the solutions obtained in (4) is −p 3 /27, deducingthat 3√ AB = −p/3.6. Prove that the possible solutions for z in (4) are given by3√A, ω3 √ A, ω 2 3√ A,3√B, ω3 √ B, ω 2 3√ Band use this result to show that the three possible solutions for y are√ω i 3 − q √√p2 + 327 + q24 + 3 ωi − q √p2 − 327 + q24 ,where i = 0, 1, 2.