So, the sign of the leading coefficient is sufficient to predict the end behavior of the function. The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph. In total we have 3 or 1 positive zeros or 2 or 0 negative zeros.The end behavior of a polynomial function is the behavior of the graph of f ( x ) as x approaches positive infinity or negative infinity. Here we can see that we have two changes of signs, hence we have two negative zeros or less but a even number of zeros. In order to find the number of negative zeros we find f(-x) and count the number of changes in sign for the coefficients: Hence our number of positive zeros must then be either 3, or 1. Descartes' rule of signs tells us that the we then have exactly 3 real positive zeros or less but an odd number of zeros. Our variables goes from positive(1) to positive(4) to negative(-3) to positive(1) to negative(-6).īetween the first two coefficients there are no change in signs but between our second and third we have our first change, then between our third and fourth we have our second change and between our 4 th and 5 th coefficients we have a third change of coefficients. Here are the coefficients of our variable in f(x): Second we count the number of changes in sign for the coefficients of f(x). Our function is arranged in descending powers of the variable, if it was not in this order we would have to rearrange the terms as our first step. We will show how it works with an example.ĭetermine the number of positive and negative real zeros for the given function (this example is also shown in our video lesson): The number of negative real zeros of the f(x) is the same as the number of changes in sign of the coefficients of the terms of f(-x) or less than this by an even number. It tells us that the number of positive real zeros in a polynomial function f(x) is the same or less than by an even numbers as the number of changes in the sign of the coefficients. Descartes' rule of sign is used to determine the number of real zeros of a polynomial function.

Is x5 x4 odd or even function