What are all the zeroes of #f(x) = 2x^3 - 5x^2 + 3x - 1#?

Question

Serenity Johnson · Accepted Answer

The Real root of #f(x) = 0# is:

#x_1 = 1/6 (5 + root(3)(44+3sqrt(177))+root(3)(44-3sqrt(177)))#

and Complex roots as below...

Given: #f(x) = 2x^3-5x^2+3x-1# First substitute #t = x-5/6# Then: #2t^3 = 2(x-5/6)^3 = 2(x^3-5/2x^2+25/12x-125/216)# #=2x^3-5x^2+25/6x-125/108# #color(white)()# #-7/6t = -7/6x+35/36 = -7/6x+105/108# #color(white)()# #2t^3-7/6t = 2x^3-5x^2+3x-5/27# So: #2t^3-7/6t-22/27 = 2x^3-5x^2+3x-1 = f(x)# Multiply through by #54# to get integer coefficients: #108t^3-63t-44 = 54 f(x)# So we want to solve: #108t^3-63t-44 = 0# Use Cardano's method, letting #t = u + v# #0 = 108(u+v)^3-63(u+v)-44# #= 108u^3+108v^3+(324uv-63)(u+v)-44# #= 108u^3+108v^3+9(36uv-7)(u+v)-44# Next make the coefficient of the #(u+v)# term zero by adding the constraint: #36uv-7 = 0#, that is #v = 7/(36u)# #= 108u^3+108(7/(36u))^3-44# #= 108u^3+343/(432u^3)-44# Multiply through by #432 u^3# to get a quadratic in #u^3#: #46656(u^3)^2-19008(u^3)+343 = 0# Then using the quadratic formula: #u^3 = (19008+-sqrt(19008^2-4*46656*343))/(2*46656)# #= (19008+-sqrt(361304064-64012032))/93312# #= (19008+-sqrt(297292032))/93312# #= (19008+-1296 sqrt(177))/93312# #= (44+-3sqrt(177))/216# Since the derivation was symmetric in #u# and #v#, one of these roots is suitable for #u^3# and the other for #v^3#. Hence the Real root is: #t = root(3)((44+3sqrt(177))/216)+root(3)((44-3sqrt(177))/216)# #= 1/6 (root(3)(44+3sqrt(177))+root(3)(44-3sqrt(177)))# and hence: #x_1 = 5/6 + t = 1/6 (5 + root(3)(44+3sqrt(177))+root(3)(44-3sqrt(177)))# The Complex roots are: #x_2 = 1/6 (5 + omega root(3)(44+3sqrt(177))+omega^2 root(3)(44-3sqrt(177)))# #x_3 = 1/6 (5 + omega^2 root(3)(44+3sqrt(177))+omega root(3)(44-3sqrt(177)))# where #omega = -1/2 +sqrt(3)/2 i# is the primitive Complex cube root of #1#.

Cora Alexander · Answer

undefined To find the zeroes of the function f(x) = 2x^3 - 5x^2 + 3x - 1, we set the function equal to zero and solve for x. By using methods such as factoring, synthetic division, or the rational root theorem, we can determine the roots of the polynomial equation. After finding the roots, we verify them by substituting them back into the original equation to ensure they satisfy f(x) = 0.