How do you find all the zeros of #(3x^6+x^2-4)(x^3+6x+7)#?

Question

Lincoln Anderson · Accepted Answer

This product has zeros:

#1#
#-1# (with multiplicity #2#)
#+-(sqrt((4sqrt(3)-3)/12))+-(sqrt((4sqrt(3)+3)/12))i#
#1/2+-(3sqrt(3))/2 i# The zeros of #(3x^6+x^2-4)(x^3+6x+7)# are all the values of #x# which are zeros of #(3x^6+x^2-4)# or #(x^3+6x+7)#. #color(white)()# Zeros of #bb(3x^6+x^2-4)# Note that all of the degrees are even and the sum of the coefficients is #0#. Hence this sextic has zeros #+-1# and quadratic factor: #(x-1)(x+1) = x^2-1# We find: #3x^6+x^2-4 = (x^2-1)(3x^4+3x^2+4)# Treating the remaining quartic factor as a quadratic in #x^2# and using the quadratic formula, we find: #x^2 = (-3+-sqrt(3^2-4(3)(4)))/(2*3)# #=(-3+-sqrt(9-48))/6# #=(-3+-sqrt(-39))/6# #=-1/2+-sqrt(39)/6i# The square roots of #a+-bi# are: #+-(sqrt((sqrt(a^2+b^2)+a)/2)) +- (sqrt((sqrt(a^2+b^2)-a)/2))i# (see https://tutor.hix.ai) So, putting #a=-1/2# and #b=sqrt(39)/6# we find: #a^2+b^2 = (-1/2)^2+(sqrt(39)/6)^2 = 1/4+39/36 = 48/36 = 4/3# So: #sqrt(a^2+b^2) = sqrt(4/3) = sqrt(4/9*3) = (2sqrt(3))/3# Hence: #x = +-(sqrt(((2sqrt(3))/3-1/2)/2))+-(sqrt(((2sqrt(3))/3+1/2)/2))i# #=+-(sqrt((4sqrt(3)-3)/12))+-(sqrt((4sqrt(3)+3)/12))i# #color(white)()# Zeros of #bb(x^3+6x+7)# Notice that #-1# is a zero, so #(x+1)# is a factor and we find: #x^3+6x+7 = (x+1)(x^2-x+7)# The remaining quadratic factor has zeros given by the quadratic formula: #x = (1+-sqrt((-1)^2-4(1)(7)))/(2*1)# #=(1+-sqrt(-27))/2# #=1/2+-(3sqrt(3))/2 i#

Anna Allen · Answer

undefined To find all the zeros of the expression (3x^6 + x^2 - 4)(x^3 + 6x + 7), you first need to find the zeros of each factor separately, as the zeros of the product are the combined zeros of the individual factors.

1. For the first factor 3x^6 + x^2 - 4, you can factorize it further if possible and find its zeros by setting it equal to zero and solving for x.

2. For the second factor x^3 + 6x + 7, you can also find its zeros by setting it equal to zero and solving for x.

The combined set of zeros for both factors will be the zeros of the given expression.