Question

For the following exercises, use the given information about the polynomial graph to write the equation. Double zero at \(\displaystyle{x}=−{3}\) and triple zero \(\displaystyle{a}{t}{x}={0}\). Passes through the point (1, 32).

Answer 1

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 \(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}+{3}\right)}^{{{2}}}\)