Exercise 5.  Let    be a polynomial of degree n.

5 (a).  Suppose that    are all real.  Show that if    is a root of  ,  then    is also a root.  In other words, the roots must be complex conjugates, something you likely learned without proof in pre-calculus.

Solution 5 (a).

See text and/or instructor's solution manual.

Since    is a root of the polynomial  ,  we have  .

Use properties (1-12) through (1-14)  of Theorem 1.1 to show that  .

Start with  ,  then  

          

This implies   .  

If   ,  then     which in turn implies that  ,  confirming that      is also a root of   .  

This solution is complements of the authors.

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