Question

If f is a polynomial of degree 3 or more how would you use real zeros of f to determine the open intervals over which f(x) > 0 or f(x) < 0. How and where does the sign of f change.

Answer 1

The sign of a graph change at its zeros. It is known that between any two consecutive zeros of a polynomial, the sign of the polynomial remains the same, that is, it remains either negative or positive. The reason is that polynomials are continuous functions and have no breaks in their graphs. This means the only way they can change signs is by crossing the x-axis.

For example, consider the graph of the polynomial

Now if we want to determine the intervals over which f(x) is positive or negative, we have to keep in mind that f will either be entirely positive or entirely negative over the selected interval, so we just have to evaluate f(x) at any arbitrary point lying in this interval and determine its sign, which In turn will be the sign of f(x) in that interval.

For example, if we consider the previous polynomial . Clearly its zeros are at x=-1,1,3. thus we have three intervals namely

Now if we want to see the sign of f(x) in , we evaluate f(x) at x=-2 (say) (or you can choose any other value in this interval), then .

Thus f(x) is negative in .

similarly, you can check in other intervals too.