Answer
$P(x)=4x^3 - 18 x^2 + 14 x + 12$.
Work Step by Step
The factor theorem says that if $f(c)=0$, then $(x-c)$ is a factor of $f(x)$ and if $(x-c)$ is a factor of $f(x)$, then $f(c)=0$.
Hence, here the function is in the form:
$P(x)=(x-(-0.5))(x-2)(x-3)=ax^3 - 4.5 ax^2 + 3.5a x + 3a$
We know that the constant coefficient is $12$. Hence, we can write:
$3a=12$
$a=4$
Thus, we have:
$P(x)=4x^3 - 18 x^2 + 14 x + 12$.
