How do you find all the zeros of #f(x)= 2x^3 - 6x^2 + 7x +9#?

1 Answer
Aug 8, 2016

Answer:

Use Cardano's method to find Real zero:

#x_1 = 1/6(6+root(3)(648+6sqrt(11670))+root(3)(648-6sqrt(11670)))#

and related Complex zeros.

Explanation:

#f(x) = 2x^3-6x^2+7x+9#

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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=-6#, #c=7# and #d=9#, so we find:

#Delta = 1764-2744+7776-8748-13608 = -15560#

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=4f(x)=8x^3-24x^2+28x+36#

#=(2x-2)^3+2(2x-2)+48#

#=t^3+2t+48#

where #t=(2x-2)#

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Cardano's method

We want to solve:

#t^3+2t+48=0#

Let #t=u+v#.

Then:

#u^3+v^3+(3uv+2)(u+v)+48=0#

Add the constraint #v=-2/(3u)# to eliminate the #(u+v)# term and get:

#u^3-8/(27u^3)+48=0#

Multiply through by #27u^3# and rearrange slightly to get:

#27(u^3)^2+1296(u^3)-8=0#

Use the quadratic formula to find:

#u^3=(-1296+-sqrt((1296)^2-4(27)(-8)))/(2*27)#

#=(1296+-sqrt(1679616+864))/54#

#=(1296+-sqrt(1680480))/54#

#=(1296+-12sqrt(11670))/54#

#=(648+-6(11670))/27#

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=1/3(root(3)(648+6sqrt(11670))+root(3)(648-6sqrt(11670)))#

and related Complex roots:

#t_2=1/3(omega root(3)(648+6sqrt(11670))+omega^2 root(3)(648-6sqrt(11670)))#

#t_3=1/3(omega^2 root(3)(648+6sqrt(11670))+omega root(3)(648-6sqrt(11670)))#

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

Now #x=1/2(2+t)=1/6(6+3t)#. So the roots of our original cubic are:

#x_1 = 1/6(6+root(3)(648+6sqrt(11670))+root(3)(648-6sqrt(11670)))#

#x_2 = 1/6(6+omega root(3)(648+6sqrt(11670))+omega^2 root(3)(648-6sqrt(11670)))#

#x_3 = 1/6(6+omega^2 root(3)(648+6sqrt(11670))+omega root(3)(648-6sqrt(11670)))#