Exercise 7.  Find all the roots of the equation    given that    is a root.  

Solution 7.

See text and/or instructor's solution manual.

Since the coefficients are real, roots come in conjugates, so that   is also a root.  

Thus,  ,  and    are factors.  Hence    is a factor.  

Dividing the polynomial by    yields  ,  which can be solved with the quadratic formula.

Here  ,  ,  ,  and  .  Thus

          

Therefore, the roots of    are .

We are done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

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