Polynomials are used to represent functions that are not straight lines by including variables raised to exponents, such as x^2. These functions can be used to project or show a variety of data, including profit versus number of employees, letter grades versus number of students getting each grade and population versus resources. Finding the maximum of a polynomial helps you to determine the most efficient point. For example, if you were using a polynomial to predict the profit versus the number of employees, the maximum would tell you how many employees to hire and what your profit would be at that point.

Arrange the polynomial into the following from: ax^2 + bx + c where a, b and c are numbers. For example, if you had 5 + 12x - 3x^2, you would rearrange it to read -3x^2 + 12x + 5.

Determine whether a, the coefficient of the x^2 term, is positive or negative. If the term is positive, the maximum value will be infinity because the value will continue to grow as x increases. If it is negative, continue to step 2.

Use the formula -b/(2a) to find the x-value for the maximum. For example, if your polynomial was -3x^2 + 12x + 5, you would use -3 for a and 12 for b and get 2.

Plug the x-value found in step 3 into the original polynomial to calculate the maximum value of the polynomial. For example, if you plugged in 2 into -3x^2 + 12x + 5, you would get 17.