# Write a simplified quartic equation with integer coefficients and positive leading coefficient as small as possible, whose single roots are -1/3 and 0 and house a double root at 0.4?

Oct 28, 2017

$f \left(x\right) = 75 {x}^{4} - 35 {x}^{3} - 8 {x}^{2} + 4 x$

#### Explanation:

Let $f \left(x\right)$ be our quartic polynomial

We are told that $f \left(x\right)$ has roots $\left\{- \frac{1}{3} , 0 , + 0.4 , + 0.4\right\}$

We are also told that $f \left(x\right)$ has integer coefficients with the leading coefficient $> 0$

Given the roots, $f \left(x\right)$ will have factors of the form: $\left(x + \frac{1}{3}\right) , x , {\left(x - \frac{2}{5}\right)}^{2}$

Given that the coefficients are integer with the coefficient of ${x}^{4} > 0$

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

$= x \left(3 x + 1\right) \left(25 {x}^{2} - 20 x + 4\right)$

$= x \left(75 {x}^{3} - 35 {x}^{2} - 8 x + 4\right)$

$= 75 {x}^{4} - 35 {x}^{3} - 8 {x}^{2} + 4 x$

Since $\left\{75 , 35 , 8 , 4\right\}$ has no common factor greater than 1, $f \left(x\right)$ is in its simplest form.