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?

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Answer 1 · 2017-10-28T03:36:27

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

Let $f (x)$ $f(x)$ be our quartic polynomial

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

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

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

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

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

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

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

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

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