Answer
$f(x)=-0.5x(x+1)(x-1)^2(x-2)$ could be one of the possible solutions.
Work Step by Step
Thanks to the real zeros we see on the graph, we can find the function through its factored form:
$f(x)=a(x+1)(x-1)^2(x-2)$
That's because when any (x-r) equals zero, f(x) equals zero. The multiplicity of (x+1) and (x-2) could be any positive odd integers since the graph crosses the x-axis at (-1,0) and (2,0). On the other hand, the multiplicity of (x-1) could be any positive even integers since the graph touches the x-axis at (1,0). For simplicity, we are going to just use the multiplicities 1 and 2.
The only thing left is to find 'a'. So, using the y-intercept, we can replace the values of 'x' and f(x) and solve for 'a':
$1=a(0+1)(0-1)^2(0-2)$
$1=a(1)(1)^2(-2)$
$1=-2a$
$1/(-2)=-2a/(-2)$
$a=-0.5$
Now, we have one possible solution:
$f(x)=-0.5x(x+1)(x-1)^2(x-2)$
