I need to find two possible polynomial function, but each one must have a different degree, and they both must have these features:
- Leading Coefficient is $-2$;
- Zeros are at $-3, -1, 4$.
I know the zeros would mean that somewhere in the equation there is: $(x+3)$, $(x+1)$, and $(x-4)$. I also know $a=-2$. I just don't know how to put it all together to find the actual functions.
UPDATE I FOUND ONE: $-2X^3+26X+24$
If you mean $-3$, $-1$ and $4$ are all the zeros (including complex zeros), then by the Fundamental Theorem of Algebra your polynomial must be of the form $f(x) = -2 (x+3)^i (x+1)^j (x-4)^k$ for some positive integers $i,j,k$. That would have $f(0) = -2 \cdot 3^i \cdot (-4)^k$, so it's good you don't need a restriction on $f(0)$.
If you mean $-3$, $-1$ and $4$ are all the real zeros, then you could have $f(x) = -2 (x+3)^i (x+1)^j (x-4)^k g(x)$ where $g(x)$ is a polynomial with no real zeros. For example, you could take $g(x) = x^2 + r$ where $r > 0$. With $i=j=k=1$ you could take $r = 5/24$ to get $f(0)=5$. Or try $g(x)=x^4+5/24$ to get a polynomial of different degree.