How do you find all the zeros of #g(x)= - 2x^3+5x^2-6x-10#?

1 Answer
Aug 11, 2016

Answer:

Use Cardano's method to find Real zero:

#x_1 = 1/6(5+root(3)(685+6sqrt(13071))+root(3)(685-6sqrt(13071)))#

and related Complex zeros.

Explanation:

#g(x) = -2x^3+5x^2-6x-10#

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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=-2#, #b=5#, #c=-6# and #d=-10#, so we find:

#Delta = 900-1728+5000-10800-10800 = -17428#

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=-108g(x)=216x^3-540x^2+648x+1080#

#=(6x-5)^3+33(6x-5)+1370#

#=t^3+33t+1370#

where #t=(6x-5)#

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Cardano's method

We want to solve:

#t^3+33t+1370=0#

Let #t=u+v#.

Then:

#u^3+v^3+3(uv+11)(u+v)+1370=0#

Add the constraint #v=-11/u# to eliminate the #(u+v)# term and get:

#u^3-1331/u^3+1370=0#

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

#(u^3)^2+1370(u^3)-1331=0#

Use the quadratic formula to find:

#u^3=(-1370+-sqrt((1370)^2-4(1)(-1331)))/(2*1)#

#=(1370+-sqrt(1876900+5324))/2#

#=(1370+-sqrt(1882224))/2#

#=(1370+-12sqrt(13071))/2#

#=685+-6sqrt(13071)#

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)(685+6sqrt(13071))+root(3)(685-6sqrt(13071))#

and related Complex roots:

#t_2=omega root(3)(685+6sqrt(13071))+omega^2 root(3)(685-6sqrt(13071))#

#t_3=omega^2 root(3)(685+6sqrt(13071))+omega root(3)(685-6sqrt(13071))#

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

Now #x=1/6(5+t)#. So the zeros of our original cubic are:

#x_1 = 1/6(5+root(3)(685+6sqrt(13071))+root(3)(685-6sqrt(13071)))#

#x_2 = 1/6(5+omega root(3)(685+6sqrt(13071))+omega^2 root(3)(685-6sqrt(13071)))#

#x_3 = 1/6(5+omega^2 root(3)(685+6sqrt(13071))+omega root(3)(685-6sqrt(13071)))#