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.
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$\begingroup$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 $0$).
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