How do you factor #4x^3-2x^2-7# ?

1 Answer
George C.
Jul 10, 2017

#4x^3-2x^2-7 = 4(x-x_1)(x-x_2)(x-x_3)#

where:

#x_k = 1/6(1+omega^(k-1) root(3)(190+3sqrt(4011))+bar(omega)^(k-1) root(3)(190-3sqrt(4011)))#

where:

#omega = -1/2+sqrt(3)/2i#

Explanation:

#f(x) = 4x^3-2x^2-7#

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Discriminant

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=4#, #b=-2#, #c=0# and #d=-7#, so we find:

#Delta = 0+0-224-21168+0 = -21392#

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=54f(x)=216x^3-108x^2-378#

#=(6x-1)^3-3(6x-1)-380#

#=t^3-3t-380#

where #t=(6x-1)#

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Cardano's method

We want to solve:

#t^3-3t-380=0#

Let #t=u+v#.

Then:

#u^3+v^3+3(uv-1)(u+v)-380=0#

Add the constraint #v=1/u# to eliminate the #(u+v)# term and get:

#u^3+1/u^3-380=0#

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

#(u^3)^2-380(u^3)+1=0#

Use the quadratic formula to find:

#u^3=(380+-sqrt((-380)^2-4(1)(1)))/(2*1)#

#=(380+-sqrt(144400-4))/2#

#=(380+-sqrt(144396))/2#

#=190+-3sqrt(4011)#

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)(190+3sqrt(4011))+root(3)(190-3sqrt(4011))#

and related Complex roots:

#t_2=omega root(3)(190+3sqrt(4011))+bar(omega) root(3)(190-3sqrt(4011))#

#t_3=bar(omega) root(3)(190+3sqrt(4011))+omega root(3)(190-3sqrt(4011))#

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

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

#x_1 = 1/6(1+root(3)(190+3sqrt(4011))+root(3)(190-3sqrt(4011)))#

#x_2 = 1/6(1+omega root(3)(190+3sqrt(4011))+bar(omega) root(3)(190-3sqrt(4011)))#

#x_3 = 1/6(1+bar(omega) root(3)(190+3sqrt(4011))+omega root(3)(190-3sqrt(4011)))#

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