Step 1
Data:
x-intercept of multiplicity \(\displaystyle{2}=-{3}\)
x-intercept of multiplicity \(\displaystyle{3}={0}\)
Degree =5
Step 2
Since it is a fifth degree polynomial function with multiplicity of 2 and 3 for some zeros, its general equation becomes:
\(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}+{3}\right)}^{{{2}}}{\left({x}-{0}\right)}\)
Simplify:
\(\displaystyle{f{{\left({x}\right)}}}={a}{x}^{{{3}}}{\left({x}+{3}\right)}^{{{2}}}\)
In order to evaluate a, use the point on the graph (1,32), therefore substitute \(\displaystyle{f{{\left({1}\right)}}}={32}\) in this equation:
\(\displaystyle{32}={a}{\left({1}\right)}^{{{3}}}{\left({1}+{3}\right)}^{{{2}}}\)
Simplify:
\(\displaystyle{32}={a}{\left({1}\right)}{\left({16}\right)}={16}{a}\)
Evaluate a:
\(\displaystyle{a}={\frac{{{32}}}{{{16}}}}={2}\)
This implies that the equation of the polynomial function is \(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}+{3}\right)}^{{{2}}}\)
Answer:
The equation of the polynomial function is \(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}+{3}\right)}^{{{2}}}\)