How do you factor #108m^3 - 500#?

1 Answer
Mar 18, 2018

Answer:

#= (3m-5)(6m + 5(1+sqrt(3)i))(6m+5(1-sqrt(3)i))#

Explanation:

#108 = 4 * 27 = 4 * 3^3#
#500 = 4 * 125 = 4 * 5^3#
#"So we have"#
#4((3m)^3 - 5^3)#
#"Now we apply "a^3-b^3 = (a-b)(a^2+ab+b^2)"#
#= 4(3m - 5)(9m^2 + 15m + 25)#
#"The quadratic factor can also be factored in factors"#
#"with complex numbers as follows :"#
#"disc : "15^2 - 4*9*25 = -675 = -27*25= -27*5^2#
#=> m = (-15 pm 5 sqrt(27) i)/18#
#=> m = (-5 pm 5 sqrt(3) i)/6#
#=> m = -(5/6) (1 pm sqrt(3) i)#
#=> 9(m + (5/6)(1 + sqrt(3) i))(m + (5/6)(1 - sqrt(3) i))#
#"So we get"#
#36(3m - 5)(m + (5/6)(1+sqrt(3)i))(m+(5/6)(1-sqrt(3)i))#
#= (3m-5)(6m + 5(1+sqrt(3)i))(6m+5(1-sqrt(3)i))#