Colorful House of Math logo
Log in

How to Analyze Polynomial Functions

Let’s look at an example of analysis of a polynomial function. The method is as follows:

Rule

Analyzing Polynomial Functions

1.
Find the zeros.
2.
Find the stationary points.
3.
Find the inflection points.

Example 1

Analyze the function f(x) = x3 + 6x2 + 8x

Example of analysis of polynomial function

1.
Find the zeros of the function by setting f(x) = 0. Since this is a polynomial function of degree 3, you cannot use the quadratic formula to solve it directly. However, as f(x) does not have a constant term, we can factorize x out of the rest of the function:
f(x) = x3 + 6x2 + 8x = x (x2 + 6x + 8)

f(x) = x3 + 6x2 + 8x = x (x2 + 6x + 8)

This means that x = 0 is a zero. You can then use the quadratic formula formula on x2 + 6x + 8 to find the other zeros: x = 6 ±(62 4 1 8 2 = 6 ±4 2 = 6 ± 2 2

Thus, x = 4 or x = 2. The zeros of f(x) are thus x = 4, x = 2 and x = 0.

2.
Find the maxima and minima by setting f(x) = 0.

Find the derivative of f(x) = x3 + 6x2 + 8x:

f(x) = 3x2 + 12x + 8

Then use the quadratic formula to find the maxima and minima of f(x): x = 12 ±(122 4 1 8 2 = 12 ±112 2 6 ± 10.58 2

Thus, x 3.15 or x 0.85.

To find the points, you need to find their corresponding y-values. You find these by putting the x-values you found back into the main function f(x): y = f(3.15) 3.08 y = f(0.85) 3.08

You now need to determine which point is a maximum and which is a minimum. You do that by drawing a sign chart. Notice that the derivative can be factorized as:

f(x) = (x + 3.15)(x + 0.85)

Sign chart of a polynomial function

Sign chart of a polynomial function

From this, you can see that the maximum is situated at (3.15, 3.08) and the minimum is situated at (0.85,3.08).
3.
Find the inflection points by setting f(x) = 0.

First, you find the second derivative of f(x) by differentiating f(x) = 3x2 + 12x + 8:

f(x) = 6x + 12.

Let f(x) = 0 and solve the equation: 6x + 12 = 0 6x = 12 x = 2

Enter this x-value into the original function f(x) to find the y-coordinate for the inflection point:

f(2) = (2)3 + 6 (2)2 + 8 (2) = 8 + 24 16 = 0

f(2) = (2)3 + 6 (2)2 + 8 (2) = 8 + 24 16 = 0

The inflection point is thus (2, 0).

By making a sign chart for the second derivative, you can see where the graph of f(x) is concave and where it is convex. Notice that f(x) can be factorized as f(x) = 6(x + 2):

Sign chart of f”(x) = 6x+12.

Sign chart of f”(x) = 6x+12.

Want to know more?Sign UpIt's free!
White arrow pointing to the LeftPrevious entry
What Are Inflection Points of a Function?
Next entryWhite arrow pointing to the right
How to Analyze Logarithmic Functions