Solve by Factoring x^3+x^2-81x-81=0

$x^{3} + x^{2} - 81 x - 81 = 0$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{3} + x^{2}) - 81 x - 81 = 0$

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

$x^{2} (x + 1) - 81 (x + 1) = 0$

Step 2

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$(x + 1) (x^{2} - 81) = 0$

Step 3

Rewrite $81$ as $9^{2}$ .

$(x + 1) (x^{2} - 9^{2}) = 0$

Step 4

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 $b = 9$ .

$(x + 1) ((x + 9) (x - 9)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x + 9) (x - 9) = 0$

Step 5

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

$x + 1 = 0$

$x + 9 = 0$

$x - 9 = 0$

Step 6

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 8

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Step 9

The final solution is all the values that make $(x + 1) (x + 9) (x - 9) = 0$ true.

$x = - 1, - 9, 9$