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) = 75x^4-35x^3-8x^2+4x#

Let #f(x)# be our quartic polynomial

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

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

Given the roots, #f(x)# will have factors of the form: #(x+1/3), x, (x-2/5)^2#

Given that the coefficients are integer with the coefficient of #x^4 >0#

#f(x) = (3x+1)x(5x-2)^2#

#= x(3x+1)(25x^2-20x+4)#

#=x(75x^3 -35x^2-8x+4)#

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

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