Exercise 5.

Every quadratic polynomial has either 2 distinct real roots, one real root of multiplicity 2, or 2 complex roots. What cases can occur for a polynomial of degree 3? Give an example for each of these cases.


We know that there are 3 roots counted according to their multiplicity, and that complex roots show up in conjugate pairs. Thus, a little bit of puzzling reveals that there are four cases.

Case 1. Three real roots, each of multiplicity 1; an example is the polynomial (x-1)(x-2)(x-3).

Case 2. One real root, one complex conjugate pair; an example is the polynomial tex2html_wrap_inline30 .

Case 3. Two real roots, one of multiplicity 1, the other one of multiplicity 2; an example is tex2html_wrap_inline32 .

Case 4. One real root of multiplicity 3; an example is tex2html_wrap_inline34 .

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Helmut Knaust
Tue Jun 24 09:55:04 MDT 1997

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