Based on this information, what is the smallest number of real zeroes that this function can have?

Question

p(x) is a polynomial function of x with real coefficients and the following characteristics:

p(-3) = 4, p(0) = -2, p(1) = 1, and p(3) = -2.


Based on this information, what is the smallest number of real zeroes that this function can have?

Answer 1

3, draw a graph.

The following equation satisfies all of the above conditions,
p(x) = -1*x^5 + (229/24)*x^3 + (1/3)*x^2 - (141/24)*x -2

and clearly has at least 3 real zeros. It can't have more since it approaches positive infinity as x -> negative infinity, and approaches negative infinity as x -> positive infinity.

Answer 2

ax^2+bx+c=0
a=-1/9
b=-1/3
c=0

x1=0
x2=3

0 is the smallest

Answer 3

four