If f is a polynomial function, and f(a) and f(b) have opposite signs, what must occur between a and b. If f(a) and f(b) have the same sign, does it necessarily mean that this will not occur?

Question

1. If f is a polynomial function, and f(a) and f(b) have opposite signs, what must occur between a and b. If f(a) and f(b) have the same sign, does it necessarily mean that this will not occur? Explain your answer.

          Can the graph of a polynomial function have no x-intercepts? Explain.

           Can the graph of a polynomial function have no y-intercepts? Explain.

Answer 1

If f is a polynomial it means that f is continuous. Thus, if f(a) and f(b) have opposite signs then f must have a zero in the open interval (a,b).

If f(a) and f(b) have the same sign then it does not necessarily mean that f will not have a zero between a and b. For example take f(x)=x^2. Then f(-1)=1=f(1)>0 but 0 is between -1 and 1 and f(0)=0.