# How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 1, -1, 2, -2, 3?

Jul 5, 2016

$f \left(x\right) = {x}^{5} - 3 {x}^{4} - 5 {x}^{3} + 15 {x}^{2} + 4 x - 12$

#### Explanation:

Each zero $a$ corresponds to a linear factor $\left(x - a\right)$, so we can write:

$f \left(x\right) = \left(x - 1\right) \left(x + 1\right) \left(x - 2\right) \left(x + 2\right) \left(x - 3\right)$

$= \left({x}^{2} - 1\right) \left({x}^{2} - 4\right) \left(x - 3\right)$

$= \left({x}^{4} - 5 {x}^{2} + 4\right) \left(x - 3\right)$

$= {x}^{5} - 3 {x}^{4} - 5 {x}^{3} + 15 {x}^{2} + 4 x - 12$