iohanetc

2021-03-08

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).

BleabyinfibiaG

Skilled2021-03-09Added 118 answers

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:
$f\left(x\right)=a{(x+3)}^{2}(x-0)$
Simplify:
$f\left(x\right)=a{x}^{3}{(x+3)}^{2}$
In order to evaluate a, use the point on the graph (1,32), therefore substitute $f\left(1\right)=32$ in this equation:
$32=a{\left(1\right)}^{3}{(1+3)}^{2}$
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 $f\left(x\right)=2{x}^{3}{(x+3)}^{2}$
Answer:
The equation of the polynomial function is $f\left(x\right)=2{x}^{3}{(x+3)}^{2}$

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