Exercise 5.  Use Bombelli's technique to get all solutions to the following depressed cubics.  
5 (c).  .  

Solution 5 (c).

See text and/or instructor's solution manual.

To solve  .  

Use the formula    for solving the depressed cubic  .  
Substitute  ,  ,  and get  ,  
which simplifies to be  .  

Assume, as Bombelli did that this expression can be put in the form  ,  where u and v are integers.  
      
Next, imitate the argument in the text that leads to equations (1-4), (1-5) and (1-6) to get  .  

The only factors of 16 are , so you can deduce (explain your reasoning) that    and    solve this system.  

Thus, one solution to    is  .

Divide    by    and get  ,  
and then solve the resulting quadratic to get the remaining solutions:   and  .  

We are done.  The roots are   .

Aside.  We can let Mathematica double check our work.

An equivalent polynomial is  .

Caveat.  You might wonder.
Why use the technique of good guessing to find that    and    solve this system   ?
Why not let computer software solve the equation    for u and v ?

Clearly there is only one solution in integers.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell