Math 4800/6080. Week Five Problems
1.If p(x, y) is a quadratic polynomial and (p = 0) has a real singular
point at (0, 0), show that (p = 0) is a pair of lines. Conclude that
this is also the case for quadratic polynomials with real singularities at
points other than (0, 0).
2. Conclude the same thing for cubic polynomials with a real singular
point of multiplicity three.
3. Suppose p(x, y) is a cubic polynomial such that (p = 0) has a real
singular point at (0, 0). Prove that if (p = 0) has another singular
point, then (p = 0) contains the line through the two singular points.
4. A plane curve (p = 0) is reducible if it is a union of two or more
plane curves. It is irreducible if it is not such a union. Thus, the curve
in Problem 3. is reducible. Find an irreducible plane curve (p = 0) for
a polynomial p(x, y) of degree 4 that has two singular points.
1