  #### Exercise 4.

Factor the polynomial in Exercise 3 completely (a) over the real numbers, (b) over the complex numbers.

We already know from Exercise 3 that the polynomial has a rational zero at . Consequently, the polynomial (x+2/3) divides evenly into Using polynomial long division, we see that How can we factor ? Such a polynomial is called bi-quadratic. It can be solved by the following trick: we substitute to obtain By the guessing method, we see that its factorization is given by: Thus the roots are y=-1 and y=-2. But from this we can calculate the roots in terms of x; recall that . Consequently the roots of the bi-quadratic polynomial are all complex: and .

Over the complex numbers, we can factor the polynomial as  Over the real numbers, the polynomials and are irreducible. Thus the polynomial's factorization is:  [Back] [Exercises] [Next]
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Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Helmut Knaust
Tue Jun 24 12:46:21 MDT 1997