Answer to Question #6038 in Algebra for Jon

Question

If you are provided with a polynomial function of degree three or greater, what are some ways you could explain the behavior without graphing the function? Which of your invented methods is best? Why?

Answer 1

The best way to explain the behaviour of the polynomial function of degree three or greater is to analyze the derivatives of this function. The standard way to explore the behaviour is:

1). find the nulls of polynomial
2). study the sign of the first derivative
3). study the sign of the second derivative

Analyzing this information you'll get the intervals of increase, decrease and concavity of polynomial.
Knowing nulls of polynomial, you'll know where function is positive or negative. From the second step( studying the sign of the first derivative) you can find out what are the intervals of increase and decrease and also critical points(points which can be points of maximum or minimum). From the third step(studying the sign of the second derivative) you'll find intervals of concavity and also turning points.