Solve by Factoring 3x^3-6x^2-27x+54=0

$3 x^{3} - 6 x^{2} - 27 x + 54 = 0$

Step 1

Factor $3$ out of $3 x^{3} - 6 x^{2} - 27 x + 54$ .

Factor $3$ out of $3 x^{3}$ .

$3 (x^{3}) - 6 x^{2} - 27 x + 54 = 0$

Factor $3$ out of $- 6 x^{2}$ .

$3 (x^{3}) + 3 (- 2 x^{2}) - 27 x + 54 = 0$

Factor $3$ out of $- 27 x$ .

$3 (x^{3}) + 3 (- 2 x^{2}) + 3 (- 9 x) + 54 = 0$

Factor $3$ out of $54$ .

$3 x^{3} + 3 (- 2 x^{2}) + 3 (- 9 x) + 3 \cdot 18 = 0$

Factor $3$ out of $3 x^{3} + 3 (- 2 x^{2})$ .

$3 (x^{3} - 2 x^{2}) + 3 (- 9 x) + 3 \cdot 18 = 0$

Factor $3$ out of $3 (x^{3} - 2 x^{2}) + 3 (- 9 x)$ .

$3 (x^{3} - 2 x^{2} - 9 x) + 3 \cdot 18 = 0$

Factor $3$ out of $3 (x^{3} - 2 x^{2} - 9 x) + 3 \cdot 18$ .

$3 (x^{3} - 2 x^{2} - 9 x + 18) = 0$

Step 2

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$3 ((x^{3} - 2 x^{2}) - 9 x + 18) = 0$

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

$3 (x^{2} (x - 2) - 9 (x - 2)) = 0$

Step 3

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$3 ((x - 2) (x^{2} - 9)) = 0$

Step 4

Rewrite $9$ as $3^{2}$ .

$3 ((x - 2) (x^{2} - 3^{2})) = 0$

Step 5

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$3 ((x - 2) ((x + 3) (x - 3))) = 0$

Remove unnecessary parentheses.

$3 ((x - 2) (x + 3) (x - 3)) = 0$

Remove unnecessary parentheses.

$3 (x - 2) (x + 3) (x - 3) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 7

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 8

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 9

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 10

The final solution is all the values that make $3 (x - 2) (x + 3) (x - 3) = 0$ true.

$x = 2, - 3, 3$

Solve by Factoring 3x^3-6x^2-27x+54=0

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