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

Question
Polynomial graphs
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).

2021-03-09
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 $$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}{\left({x}+{3}\right)}^{{{2}}}$$

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