Solve by Factoring y^3+y^2=9y+9

$y^{3} + y^{2} = 9 y + 9$

Step 1

Move all the expressions to the left side of the equation.

Subtract $9 y$ from both sides of the equation.

$y^{3} + y^{2} - 9 y = 9$

Subtract $9$ from both sides of the equation.

$y^{3} + y^{2} - 9 y - 9 = 0$

Step 2

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Step 3

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

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

Step 4

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

$(y + 1) (y^{2} - 3^{2}) = 0$

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 $a = y$ and $b = 3$ .

$(y + 1) ((y + 3) (y - 3)) = 0$

Remove unnecessary parentheses.

$(y + 1) (y + 3) (y - 3) = 0$

Step 6

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

$y + 1 = 0$

$y + 3 = 0$

$y - 3 = 0$

Step 7

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

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

$y + 1 = 0$

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 8

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

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

$y + 3 = 0$

Subtract $3$ from both sides of the equation.

$y = - 3$

Step 9

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

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

$y - 3 = 0$

Add $3$ to both sides of the equation.

$y = 3$

Step 10

The final solution is all the values that make $(y + 1) (y + 3) (y - 3) = 0$ true.

$y = - 1, - 3, 3$

Solve by Factoring y^3+y^2=9y+9

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