Exercise 4.

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

Answer.

We already know from Exercise 3 that the polynomial has a rational zero at tex2html_wrap_inline51 . Consequently, the polynomial (x+2/3) divides evenly into

displaymath37

Using polynomial long division, we see that

displaymath38

How can we factor tex2html_wrap_inline55 ? Such a polynomial is called bi-quadratic. It can be solved by the following trick: we substitute tex2html_wrap_inline57 to obtain

displaymath39

By the guessing method, we see that its factorization is given by:

displaymath40

Thus the roots are y=-1 and y=-2. But from this we can calculate the roots in terms of x; recall that tex2html_wrap_inline57 . Consequently the roots of the bi-quadratic polynomial are all complex: tex2html_wrap_inline67 and tex2html_wrap_inline69 .

Over the complex numbers, we can factor the polynomial as

displaymath41

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Over the real numbers, the polynomials tex2html_wrap_inline71 and tex2html_wrap_inline73 are irreducible. Thus the polynomial's factorization is:

displaymath43

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Helmut Knaust
Tue Jun 24 12:46:21 MDT 1997

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