Answer :
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
[tex]f(x)=a(x+1)^3(x-2)(x-3)[/tex] ...(i)
The function passes through the point (1,10). Putting x=1 and f(x)=10, we get
[tex]10=a(1+1)^3(1-2)(1-3)[/tex]
[tex]10=a(2)^3(-1)(-2)[/tex]
[tex]10=16a[/tex]
[tex]\dfrac{10}{16}=a[/tex]
[tex]0.625=a[/tex]
Putting a=0.625 in (i), we get
[tex]f(x)=0.625(x+1)^3(x-2)(x-3)[/tex]
[tex]f(x)=0.625(x+1)(x+1)(x+1)(x-2)(x-3)[/tex]
Therefore, the required polynomial function is [tex]f(x)=0.625(x+1)^3(x-2)(x-3)[/tex] or it can be written as[tex]f(x)=0.625(x+1)(x+1)(x+1)(x-2)(x-3)[/tex].