Write the polynomial equation given information about a graph With the following information, I am to write the equation of the polynomial:
Degree 3, zeros at $x=-2$, $x=1$, $x=3$, y intercept: $0,-4$
I know that the answer is: $f(x)=\frac{-2}{3}(x+2)(x-1)(x-3)$
If you look at my post history you can see that I nearly always show what I've tried and where I've gotten stuck. In this case, I do not know where to start or how to approach this problem.
How can one take the given information and calculate the polynomial? Granular, baby steps preferred.

If you know some zeroes of a polynomial, say $a$, $b$ and $c$, you can write
$$
P(x)=(x-a)(x-b)(x-c)Q(x)
$$
where $Q(x)$ is a new polynomial, in fact, if you substitute, for example, $a$, you get
$$
P(a)=(a-a)(b-a)(c-a)Q(a)=0\cdot(b-a)(c-a)Q(a)=0
$$
and the same happens if you substitute $b$ or $c$.
If you know the degree of the polynomial $P$, then you can foresee the degree of the polynomial $Q$, in fact 3 degree are already taken from $(x-a)(x-b)(x-c)$, so the degree of $Q$ is three unit lesser than the degree of $P$.
In the present case $P$ is known to have degree $3$, so $Q$ should have degree $0$, i.e. it is a constant polynomial, i.e. you have
$$
P(x)=(x-a)(x-b)(x-c)k
$$
You can obtain $k$ applying the remaining request, that the intercept is $y_0$,
$$
P(0)=y_0\quad\implies\quad(-a)(-b)(-c)k=y_0\quad\implies\quad k=-\frac{y_0}{abc}
$$
(obviously this is true if all of $a,b,c$ are different from $0$).
