Any cubic equation can be written in the form x3 + bx2 + ex + d = 0 by dividing by
Chapter 42, Problem 42.17(choose chapter or problem)
Any cubic equation can be written in the form x3 + bx2 + ex + d = 0 by dividing by theleading coefficient. The following steps show how to solve the equation by radicals.(a) Substitute x = y - b 13 to get y3 + py + q = 0 for appropriate p and q. Find p and q.(b) Show that substitution of y = Z - pl(3z) leads to z3 - p 3/(27z3) + q = O.(c) Multiply through by z3 to get a quadratic equation in z3. Show that a solution for z3 isz3 = _ '!.. + J if + p32 4 27(d) The cube roots of Z3 give z. From this we can get y and then x. Carry out the processoutlined here to solve the equation x3 + 4x2 + 4x + 3 = O.(e) Solve the cubic equation in part (d) by using Theorem 43.5 to find a rational root as afirst step.
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