How do you find all the zeros of #F(X)= 3x^2 - 2x^2 + x +4#?

1 Answer
Aug 9, 2016

Answer:

Use Cardano's method to find Real zero:

#x_1 = 1/9(2+root(3)(505+135sqrt(14))+root(3)(505-135sqrt(14)))#

and related Complex zeros...

Explanation:

Assuming that there's a typo in the question...

#f(x) = 3x^3-2x^2+x+4#

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Descriminant

The discriminant #Delta# of a cubic polynomial in the form #ax^3+bx^2+cx+d# is given by the formula:

#Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#

In our example, #a=3#, #b=-2#, #c=1# and #d=4#, so we find:

#Delta = 4-12+128-3888-432 = -4200#

Since #Delta < 0# this cubic has #1# Real zero and #2# non-Real Complex zeros, which are Complex conjugates of one another.

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Tschirnhaus transformation

To make the task of solving the cubic simpler, we make the cubic simpler using a linear substitution known as a Tschirnhaus transformation.

#0=243f(x)=729x^3-486x^2+243x+972#

#=(9x-2)^3+15(9x-2)+1010#

#=t^3+15t+1010#

where #t=(9x-2)#

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Cardano's method

We want to solve:

#t^3+15t+1010=0#

Let #t=u+v#.

Then:

#u^3+v^3+3(uv+5)(u+v)+1010=0#

Add the constraint #v=-5/u# to eliminate the #(u+v)# term and get:

#u^3-125/u^3+1010=0#

Multiply through by #u^3# and rearrange slightly to get:

#(u^3)^2+1010(u^3)-125=0#

Use the quadratic formula to find:

#u^3=(-1010+-sqrt((1010)^2-4(1)(-125)))/(2*1)#

#=(1010+-sqrt(1020100+500))/2#

#=(1010+-sqrt(1020600))/2#

#=(1010+-270sqrt(14))/2#

#=505+-135sqrt(14)#

Since this is Real and the derivation is symmetric in #u# and #v#, we can use one of these roots for #u^3# and the other for #v^3# to find Real root:

#t_1=root(3)(505+135sqrt(14))+root(3)(505-135sqrt(14))#

and related Complex roots:

#t_2=omega root(3)(505+135sqrt(14))+omega^2 root(3)(505-135sqrt(14))#

#t_3=omega^2 root(3)(505+135sqrt(14))+omega root(3)(505-135sqrt(14))#

where #omega=-1/2+sqrt(3)/2i# is the primitive Complex cube root of #1#.

Now #x=1/9(2+t)#. So the zeros of our original cubic are:

#x_1 = 1/9(2+root(3)(505+135sqrt(14))+root(3)(505-135sqrt(14)))#

#x_2 = 1/9(2+omega root(3)(505+135sqrt(14))+omega^2 root(3)(505-135sqrt(14)))#

#x_3 = 1/9(2+omega^2 root(3)(505+135sqrt(14))+omega root(3)(505-135sqrt(14)))#