Question

How to find th number of real solutions for a polynomial?

Answer 1

The highest degree of the polynomial tells you the maximum number of zeroes, but doesn’t say whether there are duplicates and doesn’t say whether they are real or complex. One way to find out is to graph the polynomial. This can just be a sketch. By plugging in values you can evaluate the polynomial. Let’s say you choose two values for the independent variable: a and b. If the polynomial evaluates as positive for a and negative for b, or vice versa, it means there’s a real solution between a and b. You can do this for a series of values and look for changes from positive to negative or negative to positive. And in this way you can determine how many real solutions there are. If by chance you find a value c which makes the polynomial zero, then c is a zero and you can use synthetic or long division to divide by the zero and reduce the degree of the polynomial. Then repeat the process to find more zeroes or determine how many real solutions there are. A graph does help to see at a glance where the real solutions are, and some calculators and apps will draw the graph for you.