Question

Sketch a graph of a fifth-degree polynomial f(x) such that f(x) < 0 when x< -2 or 4 < x < 7, and f(x)>0 when -2<x<0, 0<x<4 or when x>7. What conclusions can you make about the sign of the leading coefficient and its end behaviour (the sign changes the direction of the graph)? Explain your answer using terminology.

Answer 1

Since f(x) is neither positive nor negative at x=0, that makes 0 a double root.
f(x) = x^2(x+2)(x-4)(x-7)
see the graph at

https://www.wolframalpha.com/input/?i=x%5E2%28x%2B2%29%28x-4%29%28x-7%29