 # 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). iohanetc 2021-03-08 Answered
For the following exercises, use the given information about the polynomial graph to write the equation. Double zero at $x=-3$ and triple zero $atx=0$. Passes through the point (1, 32).
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Step 1 Data: x-intercept of multiplicity $2=-3$ x-intercept of multiplicity $3=0$ 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: $f\left(x\right)=a{\left(x+3\right)}^{2}\left(x-0\right)$ Simplify: $f\left(x\right)=a{x}^{3}{\left(x+3\right)}^{2}$ In order to evaluate a, use the point on the graph (1,32), therefore substitute $f\left(1\right)=32$ in this equation: $32=a{\left(1\right)}^{3}{\left(1+3\right)}^{2}$ Simplify: $32=a\left(1\right)\left(16\right)=16a$ Evaluate a: $a=\frac{32}{16}=2$ This implies that the equation of the polynomial function is $f\left(x\right)=2{x}^{3}{\left(x+3\right)}^{2}$ Answer: The equation of the polynomial function is $f\left(x\right)=2{x}^{3}{\left(x+3\right)}^{2}$