Find the derivative of the polynomial. This is a simpler polynomial -- one degree less -- that describes how the original polynomial changes. The derivative is zero when the original polynomial is at a turning point -- the point at which the graph is neither increasing nor decreasing. The roots of the derivative are the places where the original polynomial has turning points. Because the derivative has degree one less than the original polynomial, there will be one less turning point -- at most -- than the degree of the original polynomial.
Form the derivative of a polynomial term by term. The pattern is this: bX^n becomes bnX^(n - 1). Apply the pattern to each term except the constant term. Derivatives express change and constants do not change, so the derivative of a constant is zero. For example, the derivatives of X^4 + 2X^3 - 5X^2 - 13X + 15 is 4X^3 + 6X^2 - 10X - 13. The 15 disappears because the derivative of 15, or any constant, is zero. The derivative 4X^3 + 6X^2 - 10X - 13 describes how X^4 + 2X^3 - 5X^2 - 13X + 15 changes.
Find the turning points of an example polynomial X^3 - 6X^2 + 9X - 15. First find the derivative by applying the pattern term by term to get the derivative polynomial 3X^2 -12X + 9. Set the derivative to zero and factor to find the roots. 3X^2 -12X + 9 = (3X - 3)(X - 3) = 0. This means that X = 1 and X = 3 are roots of 3X^2 -12X + 9. This means that the graph of X^3 - 6X^2 + 9X - 15 will change directions when X = 1 and when X = 3.