Question

Find all solutions to x³+3x² − 9x − 27 = 0 by factoring the equation.

Answer 1

The solutions are
`x=-3,\:x=3`

Explanation:

To factor this cubic polynomial
`x^3+3x^2-9x-27` you must:

• Group the polynomial into two sections

`x^3+3x^2-9x-27=\left(x^3+3x^2\right)+\left(-9x-27\right)`

• Factor out -9 from
  `(-9x-27)`

`(-9x-27)=-9\left(x+3\right)`

• Factor out
  `x^2` from
  `(x^3+3x^2)`

`(x^3+3x^2)=x^2\left(x+3\right)`

`x^3+3x^2-9x-27=-9\left(x+3\right)+x^2\left(x+3\right)`

• Factor out common term
  `x+3`

`-9\left(x+3\right)+x^2\left(x+3\right)=\left(x+3\right)\left(x^2-9\right)`

`x^3+3x^2-9x-27=\left(x+3\right)\left(x^2-9\right)`

• Factor
  `x^2-9`

`x^2-9=\left(x+3\right)\left(x-3\right)`

`x^3+3x^2-9x-27= \left(x+3\right)\left(x+3\right)\left(x-3\right)`

`x^3+3x^2-9x-27=\left(x+3\right)^2\left(x-3\right)=0`

• Using the Zero Factor Theorem: = 0 if and only if = 0 or = 0

`x+3=0, \quad{x=-3}\\x-3=0, \quad{x=3}`

The solutions are

`x=-3,\:x=3`