If you have an arbitrary polynomial P(x) of the n’th degree in x, and you graph the equation y = P(x), describe the overall shape of this graph. What happens when x gets large in the positive and negative directions, and how many peaks and valleys does the graph have? How many times does the graph cross the x-axis? (To simplify, let the sign of the highest power term in P(x) be positive.) Hint: See chapter 11 of the text.

Answer 1

I recommend that you read chapter 11 and take a stab at this yourself.

In general, what happens at very large values of x and -x depends upon the sign of the highest order term, x^n, and whether n is evn or odd. The higher the value of n, the more times the function goes up and down, but the number of maxima, minima and zeroes will depend upon the coefficients of all terms. An nth order polynomial can have n roots, but some may be equal and some may be complex numbers. Complex roots come in conjuagate pairs.