# Question #e1c6e

May 25, 2017

If $z$ is a zero of polynomial $P$, then $x - z$ is a factor of $P$.

So the factors of the desired polynomial must include

$\left(x + 6\right) \left(x + 1\right) x \left(x - 1\right) \left(x - 6\right)$.

We can put a constant factor in as well.
If the list provided only names and does not count zeros, then we could also raise the factors given to powers greater than $1$.

If you are required to write the polynomial in standard form, I think you'll find it less cumbersome to multiply in a different order than the one above.

$\left(x + 6\right) \left(x + 1\right) x \left(x - 1\right) \left(x - 6\right) = x \left[\left(x + 1\right) \left(x - 1\right)\right] \left[\left(x + 6\right) \left(x - 6\right)\right]$.

$= x \left({x}^{2} - 1\right) \left({x}^{2} - 36\right)$

$= 1 \left({x}^{4} - 37 {x}^{2} + 36\right)$

$= {x}^{5} - 27 {x}^{3} + 36 x$