How to Find Turning Points of a Polynomial

By Carlos Mano
Image of a student in algebra class.

A polynomial is an expression that deals with decreasing powers of ‘x’, such as in this example: 2X^3 + 3X^2 - X + 6. When a polynomial of degree two or higher is graphed, it produces a curve. This curve may change direction, where it starts off as a rising curve, then reaches a high point where it changes direction and becomes a downward curve. Conversely, the curve may decrease to a low point at which point it reverses direction and becomes a rising curve. If the degree is high enough, there may be several of these turning points. There can be as many turning points as one less than the degree -- the size of the largest exponent -- of the polynomial.

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.


It will save a lot of time if you factor out common terms before starting the search for turning points. For example. the polynomial 3X^2 -12X + 9 has exactly the same roots as X^2 - 4X + 3. Factoring out the 3 simplifies everything.


The degree of the derivative gives the maximum number of roots. In the case of multiple roots or complex roots, the derivative set to zero may have fewer roots, which means the original polynomial may not change directions as many times as you might expect. For example, the equation Y = (X - 1)^3 does not have any turning points.