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

1 Answer

75x^4-35x^3-8x^2+4x=0

Explanation:

We have roots of:

x=-1/3, 0, 2/5, 2/5

We can then say:

x+1/3=0, x=0, x-2/5=0, x-2/5=0

And then:

(x+1/3)(x)(x-2/5)(x-2/5)=0

And now starts the multiplying:

(x^2+1/3x)(x-2/5)(x-2/5)=0

(x^2+1/3x)(x^2-4/5x+4/25)=0

x^4+1/3x^3-4/5x^3-4/15x^2+4/25x^2+4/75x=0

75x^4+25x^3-60x^3-20x^2+12x^2+4x=0

75x^4-35x^3-8x^2+4x=0