How do you factor and solve #x^3-216=0#?

2 Answers
Apr 2, 2018

Answer:

#=> x = { 6 , -3 pm 3sqrt(3)i } #

Explanation:

#x^3 = 216 => x = 6 #

#=> (x-6)# is a factor

#(x-6)(x^2 +bx + 36 ) = x^3 - 216 #

#=> -6x^2 +bx^2 = 0x^2 #

#=> b-6 = 0 => b = 6 #

#=> (x-6)(x^2 + 6x + 36 ) = 0 #

#=> x = { 6 , -3 pm 3sqrt(3)i } #

Using quadratic formula

Answer:

See below:

Explanation:

According to the fundamental theorem of Algebra, a polynomial to the degree #n# will have #n# roots. In our case, a cubic will have three roots.

#x=6# is the first solution, which is obtained by solving the equation (taking the cube root of #216#).

If# x=6# is a solution, then #(x-6)# must be a factor of this polynomial. So, if you divide this polynomial by #(x-6)# using algebraic division, you will get the quadratic:

#x^2+6x+36#

Solving this quadratic will give you the remaining two roots:

#x^2+6x+36=0#

#(x+3)^2-9+36=0#

#(x+3)^2+27=0#

#(x+3)= sqrt(-27)#

#(x+3)= ±3sqrt3i#

#x= -3±3sqrt3i#

The catch here is to remember that if you take the square root of any number you get two answers(#±# the answer). I solved this quadratic by 'completing the square', there is also an alternative quadratic formula which can be used to solve quadratics. The non-real solutions exist in conjugate pairs #(a±bi)#.

Therefore, the solutions are:

#x= 6, (-3+sqrt3i), (-3-sqrt3i)#