Data:
x — intercept=-2,1,3

© — intercept of multiplicity =-4

Degree=4

Since it is a third degree function with three x intercepts, its general equation becomes: \(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}+{2}\right)}{\left({x}—{1}\right)}{\left({x}-{3}\right)}\)

In order to evaluate a, use the y - intercept (0,-4), therefore substitute f(0)=-4 in this equation:

\(\displaystyle-{4}={a}{\left({0}+{2}\right)}{\left({0}—{1}\right)}{\left({0}-{3}\right)}\)

Simplify: -4=6a

Evaluate a: \(\displaystyle{a}=-\frac{{4}}{{6}}=-{\left(\frac{{2}}{{3}}\right)}\)

This implies that the equation of the given polynomial function is f(x) =

\(\displaystyle{\left(-{\left(\frac{{2}}{{3}}\right)}\right)}{\left({x}+{3}\right)}{)}{\left({x}—{1}\right)}{\left({x}-{3}\right)}\)

© — intercept of multiplicity =-4

Degree=4

Since it is a third degree function with three x intercepts, its general equation becomes: \(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}+{2}\right)}{\left({x}—{1}\right)}{\left({x}-{3}\right)}\)

In order to evaluate a, use the y - intercept (0,-4), therefore substitute f(0)=-4 in this equation:

\(\displaystyle-{4}={a}{\left({0}+{2}\right)}{\left({0}—{1}\right)}{\left({0}-{3}\right)}\)

Simplify: -4=6a

Evaluate a: \(\displaystyle{a}=-\frac{{4}}{{6}}=-{\left(\frac{{2}}{{3}}\right)}\)

This implies that the equation of the given polynomial function is f(x) =

\(\displaystyle{\left(-{\left(\frac{{2}}{{3}}\right)}\right)}{\left({x}+{3}\right)}{)}{\left({x}—{1}\right)}{\left({x}-{3}\right)}\)