"First we search the zeros"
x^5 + 3 x^2 - x = x(x^4 + 3 x - 1)
x^4 + 3 x - 1 = (x^2 + a x + b)(x^2 - a x + c)
=> b+c-a^2 = 0 ,
" "a(c-b) = 3 ,
" "bc = -1
=> b+c = a^2, " "c-b = 3/a
=> 2c = a^2+3/a, " "2b = a^2-3/a
=> 4bc = a^4 - 9/a^2 = -4
"Name k = a²"
"Then we get the following cubic equation"
k^3 + 4 k - 9 = 0
"Substitute k = r p :"
r^3 p^3 + 4 r p - 9 = 0
=> p^3 + (4/r^2) p - 9/r^3 = 0
"Choose r so that 4/r² = 3 => r = "2/sqrt(3)
"Then we get"
=> p^3 + 3 p - (27/8) sqrt(3) = 0
"Substitute p = t - 1/t :"
=> t^3 - 1/t^3 - (27/8) sqrt(3) = 0
=> t^6 - (27/8) sqrt(3) t^3 - 1 = 0
"Substitute u = t³, then we get a quadratic equation"
=> u^2 - (27/8) sqrt(3) u - 1 = 0
disc : 3*(27/8)^2 + 4 = 2443/64
=> u = ((27/8) sqrt(3) pm sqrt(2443)/8)/2
=> u = (27 sqrt(3) pm sqrt(2443))/16
"Take the solution with the + sign : "
u = 6.0120053
=> t = 1.8183317
=> p = 1.2683771
=> k = 1.4645957
=> a = 1.2102048
=> b = -0.50716177
=> c = 1.9717575
x^4 + 3 x - 1 = (x^2 + a x + b)(x^2 - a x + c)
"So the roots are"
x = (-a pm sqrt(a^2-4*b))/2
=> x = -0.6051024 pm 0.93451094
=> x = -1.53961334 " OR " 0.32940854
"And"
x = (a pm sqrt(a^2-4*c))/2
=> x = "not real as " a^2-4*c < 0
"So we have three zeros for our original quintic equation :"
x = = -1.53961334 " OR " 0 " OR " 0.32940854
"The end behavior is"
lim_{x->-oo} = -oo" , and"
lim_{x->+oo} = +oo."
"So we have"
-oo "........." -1.53961334 "........." 0 ".........." 0.32940854 "........" +oo
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