Factor over the Complex Numbers x^3-3x^2+81x-243

$x^{3} - 3 x^{2} + 81 x - 243$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{3} - 3 x^{2}) + 81 x - 243$

Factor out the greatest common factor (GCF) from each group.

$x^{2} (x - 3) + 81 (x - 3)$

Step 2

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$(x - 3) (x^{2} + 81)$

Step 3

Multiply the constant in the polynomial $x^{2} + 81$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$(x - 3) (x^{2} - 81 i^{2})$

Step 4

Rewrite $81 i^{2}$ as ${(9 i)}^{2}$ .

$(x - 3) (x^{2} - {(9 i)}^{2})$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = x$ and $i = 9 i$ .

$(x - 3) ((x + 9 i) (x - (9 i)))$

Step 6

Factor.

Multiply $9$ by $- 1$ .

$(x - 3) ((x + 9 i) (x - 9 i))$

Remove unnecessary parentheses.

$(x - 3) (x + 9 i) (x - 9 i)$