A polynomial is a function that can be written in the form ax^n+bx^(n-1)+..+c, where n≥0. Complex numbers have negative exponents.

Can a polynomial also have a negative exponent?

The answer is yes! Let’s use an example to demonstrate this. If you were to graph y=x^2-3x, you would see that it has two zeroes and one maximum value (which happens to be at x=-1).

statistic, analytic, diagram @ Pixabay

If we were to graph y=x^(-n+m), where n and m are integers, you would again see that it has two zeroes (those of the power) and one maximum value. A polynomial can have a negative exponent, as long as the base is an integer! 

The first thing I did was create some basic functions in my calculator: f(x)=a*x^b; g(x)=-ax^c; h(x)=(-ax)*g(y); i=(-ax)*h() .. MATH PLEASE LOG OUT OF YOUR ACCOUNT BEFORE READING FURTHER FOR PRIVACY PURPOSES.

Next, I plotted them on a graph to see if they had the same shape. When I plotted them on a graph, it was obvious that f(x) and g(x) have different shapes because of their negative exponents! In Graph A you can see where x=-a/b is one zero (the minimum value).

It also has another point at (-c-d)/(-e+f), which is not shown in this image. That’s an additional zeroes for “g”. In order to plot y=h() or i(), we need two functions with nonnegative coefficients; then we would be able to draw more lines from each other. 


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