Question

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

Answer 1

`= (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))`