# How do you find a polynomial with roots 4, −5, and 7?

Jul 30, 2016

$f \left(x\right) = {x}^{3} - 6 {x}^{2} - 27 x + 140$
Each zero $a$ corresponds to a linear factor $\left(x - a\right)$, so a suitable polynomial would be:
$f \left(x\right) = \left(x - 4\right) \left(x + 5\right) \left(x - 7\right) = {x}^{3} - 6 {x}^{2} - 27 x + 140$
Any polynomial in $x$ with these zeros will be a multiple (scalar or polynomial) of this $f \left(x\right)$.