Question

The degree of the polynomial function f(x) is 4 The roots of the equation f(x)=0 are -4,-1, 2 and 5 Which graph could be the graph of f(x)

Answer 1

If the roots of the equation f(x)=0 are -4,-1, 2 and 5, then binomials (x+4), (x+1), (x-2) and (x-5) are factors of needed polynomial.

Thus, the polynomial will have form

$f(x)=a(x+4)(x+1)(x-2)(x-5),$ where a is real coefficient (positive or negative). This polynomial has degree 4 as needed.

The form of the graph depends on the sign of coefficient a. Attached diagrams show two different cases of possible forms of graphs (first one for positive coefficient a, second one for negative coefficient a).