For a polynomial function of the fifth degree, determine the polynomial function in the form f(x)=a(x-m)(x-n)(x-p)(x-q)(x-r) using the following graph

Given:

The graph of a polynomial.

To find:

The polynomial function for the given graph.

Solution:

If the graph of function intersect the x-axis at x=c, then (x-c) is a factor of that function f(x).

From the given graph it is clear that the graph of the polynomial function intersect the x-axis at x=-1, x=2, x=3.

So, (x+1), (x-2) and (x-3) are the factors of required polynomial.

At x=2 and x=3, it look like a linear function. So, the multiplicity of (x-2) and (x-3) are 1.

At x=-1, it look like a cubic function. So, the multiplicity of (x+1) is 3.

The required polynomial is

         ...(i)

The function passes through the point (1,10). Putting x=1 and f(x)=10, we get

Putting a=0.625 in (i), we get

Therefore, the required polynomial function is  or it can be written as.