What is the solution set for #3x^5-48x=0#?

1 Answer
Aug 8, 2015

Answer:

# 0, +-2, +-2i #

Explanation:

Note that this is polynomial an equation of the 5th degree, so it should have 5 solutions.

# 3x^5 - 48x = 0 #
# => 3x (x^4 - 16) = 0 #
# => x ((x^2)^2 - 4^2) = 0 # (Dividing both sides by 3)
# => x (x^2 + 4) (x^2 - 4) = 0 # (Since # x^2 - y^2 = (x + y)(x - y) #)
# => x (x^2 - (-4)) (x^2 - 4) = 0 # (*)
# => x (x^2 - (-4)) (x^2 - 4) = 0 #
# => x (x^2 - (2i)^2) (x^2 - 2^2) = 0 # ( # i^2 = -1 #)
# => x (x + 2i) (x - 2i) (x + 2) (x - 2) = 0 #
# => x = 0, +-2, +-2i #

If you are not looking for complex roots, at the step marked (*), note that #x^2 + 4# is always positive for all real values of #x#, and thus divide by #x^2 + 4#. Then you can continue in the exact same way as given.