Abstract Algebra: Theory and Applications

Complete the square to solve the general quadratic equation

to obtain

The discriminant of the quadratic equation $\mathrm{\Delta }=b^2-4ac$ determines the nature of the solutions of the equation. If $\mathrm{\Delta }>0\text{,}$ the equation has two distinct real solutions. If $\mathrm{\Delta }=0\text{,}$ the equation has a single repeated real root. If $\mathrm{\Delta }<0\text{,}$ there are two distinct imaginary solutions.

Show that any cubic equation of the form

can be reduced to the form $y^3+py+q=0$ by making the substitution $x=y-b/3\text{.}$

Prove that the cube roots of 1 are given by

Make the substitution

for $y$ in the equation $y^3+py+q=0$ and obtain two solutions $A$ and $B$ for $z^3\text{.}$

Prove that the possible solutions for $z$ in (4) are given by

and use this result to show that the three possible solutions for $y$ are

where $i=0,1,2\text{.}$

The discriminant of the cubic equation is

Show that $y^3+py+q=0$

Solve the following cubic equations.

Show that the general quartic equation

can be reduced to

by using the substitution $x=y-a/4\text{.}$

Show that

Show that the right-hand side of Exercise 17.6.10 can be put in the form $(my+k{)}^2$ if and only if

From Exercise 17.6.11 obtain the resolvent cubic equation

Solving the resolvent cubic equation, put the equation found in Exercise 17.6.10 in the form

to obtain the solution of the quartic equation.

Use this method to solve the following quartic equations.