Solve by Factoring x^3+3x^2-49x-147=0

$x^{3} + 3 x^{2} - 49 x - 147 = 0$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{3} + 3 x^{2}) - 49 x - 147 = 0$

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

$x^{2} (x + 3) - 49 (x + 3) = 0$

Step 2

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

$(x + 3) (x^{2} - 49) = 0$

Step 3

Rewrite $49$ as $7^{2}$ .

$(x + 3) (x^{2} - 7^{2}) = 0$

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 $b = 7$ .

$(x + 3) ((x + 7) (x - 7)) = 0$

Remove unnecessary parentheses.

$(x + 3) (x + 7) (x - 7) = 0$

Step 5

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

$x + 3 = 0$

$x + 7 = 0$

$x - 7 = 0$

Step 6

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 7

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 8

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 9

The final solution is all the values that make $(x + 3) (x + 7) (x - 7) = 0$ true.

$x = - 3, - 7, 7$