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 .

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

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

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*Helmut Knaust *

Tue Jun 24 09:55:04 MDT 1997

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