Question

How do you solve `27x^3-1=0` by factoring?

Answer 1

`x=1/3` is the only solution in `RR`

Explanation:

Solving this eqation is determined by factorizing the binomial `27x^3-1` using the following polynomial identity:

`color(red)(a^3-b^3=(a-b)(a^2+ab+b^2))`

`27x^3-1=0`

`rArr3^3xxx^3-1^3=0`

`rArr(3x)^3-1^3=0`

`rArrcolor(red)((3x-1)((3x)^2+(3x)(1)+1^2))`

`rArr(3x-1)(9x^2+3x+1)=0`

`rArrcolor(blue)((3x-1)=0`
`rArr3x=1`
`rArrx=1/3`

OR
`color(blue)(9x^2+3x+1=0`

Let us determine the solution by applying quadratic formula
`delta=b^2-4ac=3^2-4(9)(1)`
`delta=9-36=-27`

`delta<0` No solution in `RR` for `color(blue)(9x^2+3x+1=0`
Therefore,

`x=1/3` is the only solution in `RR` for the given equation