How do you find all the zeros of #f(x)=x^3+x^2-11x-30#?

1 Answer
Aug 12, 2016

Answer:

Use Cardano's method to find Real zero:

#x_1 = 1/3(-1+root(3)((-709+9sqrt(4265))/2)+root(3)((-709-9sqrt(4265))/2))#

and related Complex zeros.

Explanation:

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#f(x) = x^3+x^2-11x-30#

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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=1#, #b=1#, #c=-11# and #d=-30#, so we find:

#Delta = 121+5324+120-24300+5940 = -12795#

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=27f(x)=27x^3+27x^2-297x-810#

#=(3x+1)^3-102(3x+1)-709#

#=t^3-102t-709#

where #t=(3x+1)#

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Cardano's method

We want to solve:

#t^3-102t-709=0#

Let #t=u+v#.

Then:

#u^3+v^3+3(uv-34)(u+v)-709=0#

Add the constraint #v=34/u# to eliminate the #(u+v)# term and get:

#u^3+39304/u^3-709=0#

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

#(u^3)^2-709(u^3)+39304=0#

Use the quadratic formula to find:

#u^3=(709+-sqrt((-709)^2-4(1)(39304)))/(2*1)#

#=(-709+-sqrt(502681-157216))/2#

#=(-709+-sqrt(345465))/2#

#=(-709+-9sqrt(4265))/2#

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)((-709+9sqrt(4265))/2)+root(3)((-709-9sqrt(4265))/2)#

and related Complex roots:

#t_2=omega root(3)((-709+9sqrt(4265))/2)+omega^2 root(3)((-709-9sqrt(4265))/2)#

#t_3=omega^2 root(3)((-709+9sqrt(4265))/2)+omega root(3)((-709-9sqrt(4265))/2)#

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

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

#x_1 = 1/3(-1+root(3)((-709+9sqrt(4265))/2)+root(3)((-709-9sqrt(4265))/2))#

#x_2 = 1/3(-1+omega root(3)((-709+9sqrt(4265))/2)+omega^2 root(3)((-709-9sqrt(4265))/2))#

#x_3 = 1/3(-1+omega^2 root(3)((-709+9sqrt(4265))/2)+omega root(3)((-709-9sqrt(4265))/2))#