# How do you find a polynomial function that has zeros x=-3, 1, 5, 6 and degree n=5?

Feb 10, 2017

$= {x}^{5} - 6 {x}^{4} - 22 {x}^{3} + 108 {x}^{2} + 189 x - 270$

#### Explanation:

The function could be the following:

$f \left(x\right) = {\left(x + 3\right)}^{2} \left(x - 1\right) \left(x - 5\right) \left(x - 6\right)$

that's

$= \left({x}^{2} + 6 x + 9\right) \left({x}^{2} - 6 x + 5\right) \left(x - 6\right)$

$= \left({x}^{4} \cancel{- 6 {x}^{3}} + 5 {x}^{2} \cancel{+ 6 {x}^{3}} - 36 {x}^{2} + 30 x + 9 {x}^{2} - 54 x + 45\right) \left(x - 6\right)$

$= \left({x}^{4} - 22 {x}^{2} - 24 x + 45\right) \left(x - 6\right)$

$= {x}^{5} - 6 {x}^{4} - 22 {x}^{3} + 132 {x}^{2} - 24 {x}^{2} + 144 x + 45 x - 270$

$= {x}^{5} - 6 {x}^{4} - 22 {x}^{3} + 108 {x}^{2} + 189 x - 270$