How do you find all the zeros of #g(x) = 9x^3- 7x^2 + 10x - 4#?

1 Answer
Aug 11, 2016

Answer:

Use Cardano's method to find Real zero:

#x_1 = 1/27(7+root(3)(-1882+81sqrt(2185))+root(3)(-1882-81sqrt(2185)))#

and related Complex zeros.

Explanation:

#g(x) = 9x^3-7x^2+10x-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=9#, #b=-7#, #c=10# and #d=-4#, so we find:

#Delta = 4900-36000-5488-34992+45360 = -26220#

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=2187g(x)=19683x^3-15309x^2+21870x-8748#

#=(27x-7)^3+663(27x-7)-3764#

#=t^3+663t-3764#

where #t=(27x-7)#

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Cardano's method

We want to solve:

#t^3+663t-3764=0#

Let #t=u+v#.

Then:

#u^3+v^3+3(uv+221)(u+v)-3764=0#

Add the constraint #v=-221/u# to eliminate the #(u+v)# term and get:

#u^3-10793861/u^3-3764=0#

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

#(u^3)^2-3764(u^3)-10793861=0#

Use the quadratic formula to find:

#u^3=(3764+-sqrt((-3764)^2-4(1)(-10793861)))/(2*1)#

#=(-3764+-sqrt(14167696+43175444))/2#

#=(-3764+-sqrt(57343140))/2#

#=(-3764+-162(2185))/2#

#=-1882+-81(2185)#

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)(-1882+81sqrt(2185))+root(3)(-1882-81sqrt(2185))#

and related Complex roots:

#t_2=omega root(3)(-1882+81sqrt(2185))+omega^2 root(3)(-1882-81sqrt(2185))#

#t_3=omega^2 root(3)(-1882+81sqrt(2185))+omega root(3)(-1882-81sqrt(2185))#

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

Now #x=1/27(7+t)#. So the zeros of our original cubic are:

#x_1 = 1/27(7+root(3)(-1882+81sqrt(2185))+root(3)(-1882-81sqrt(2185)))#

#x_2 = 1/27(7+omega root(3)(-1882+81sqrt(2185))+omega^2 root(3)(-1882-81sqrt(2185)))#

#x_3 = 1/27(7+omega^2 root(3)(-1882+81sqrt(2185))+omega root(3)(-1882-81sqrt(2185)))#