# How do you find all zeros of f(x)=x^3-4x^2-25x+100?

Jan 7, 2017

$x = 5$, $x = - 5$ or $x = 4$

#### Explanation:

Note that the ratio of the first and second terms is the same as that of the third and fourth terms, so this cubic factors by grouping and we find:

$0 = {x}^{3} - 4 {x}^{2} - 25 x + 100$

$\textcolor{w h i t e}{0} = \left({x}^{3} - 4 {x}^{2}\right) - \left(25 x - 100\right)$

$\textcolor{w h i t e}{0} = {x}^{2} \left(x - 4\right) - 25 \left(x - 4\right)$

$\textcolor{w h i t e}{0} = \left({x}^{2} - 25\right) \left(x - 4\right)$

$\textcolor{w h i t e}{0} = \left({x}^{2} - {5}^{2}\right) \left(x - 4\right)$

$\textcolor{w h i t e}{0} = \left(x - 5\right) \left(x + 5\right) \left(x - 4\right)$

Hence $x = 5$, $x = - 5$ or $x = 4$