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The definition of polynomial can be extended to more than one variable, yielding a different way to think about some implicitly defined equations. For example, f((x,y)) = x^2 + y^2 - 1 can be thought of as a polynomial in two variables, x and y.
1.1 An ordered pair of real numbers (a,b) is a root of a polynomial f if f((a,b)) = 0. Defining f as in the example, plot f((x,y)) = 0 on a graphing calculator. Under our new interpretation of this equation, what does the plot depict? Besides the appearance of the plot, in what dramatic way does this situation differ from when our polynomials are in only one variable?
1.2 Given a polynomial g, we are often interested in when the plot of g((x,y)) = 0 looks "smooth" at some root (a,b). We can use calculus to check for this. (a) Let g((x,y)) = x^3 + y^3 - 6xy, the folium of Descartes discussed in class. (Plot g((x, y)) = 0). First, treating y like a constant and x like a variable, compute the "derivative" of g with respect to x.
(b) Instead, treat x like a constant and y like a variable and compute the "derivative" of g with respect to y.
(c) Evaluate the "derivatives" you obtained in (a) and (b) at the root (0,0). How do they compare here? What about at (3,3)?
8uipoxdoYBodoY(0=(fx))y O[d)izx-gx-zf=((hx))y I(p) calculations for h, but instead consider the roots (0,0) and (-1,0). From this, form a conjecture about what must be true of the "derivatives" of a polynomial at a root (a, b) in order to appear non-"smooth".
