Question

In Problems 65–68, construct a polynomial function that might have the given graph. (More than one answer may be possible.)

Short answer

The possible function is $f\left(x\right)=x{(x-1)}^2\left(x-2\right)$

Step-by-step solution

Step 1. Given information 

The graph is

Step 2. To find the possible function from graph. 

In the given graph we can see that there are 3 zeros because the graph is crossing the x-axis at

$x=0,2$, so these zeros have odd multiplicity

and touches x-axis at $x=1$

so it has even multiplicity.

So the factors of the function are: $x,{\left(x-1\right)}^2,\left(x-2\right)$

Hence the function is

$f\left(x\right)=x{(x-1)}^2\left(x-2\right)$