Solve by Factoring 2y^3-13y^2-7y=0

$2 y^{3} - 13 y^{2} - 7 y = 0$

Step 1

Factor $y$ out of $2 y^{3} - 13 y^{2} - 7 y$ .

Factor $y$ out of $2 y^{3}$ .

$y (2 y^{2}) - 13 y^{2} - 7 y = 0$

Factor $y$ out of $- 13 y^{2}$ .

$y (2 y^{2}) + y (- 13 y) - 7 y = 0$

Factor $y$ out of $- 7 y$ .

$y (2 y^{2}) + y (- 13 y) + y \cdot - 7 = 0$

Factor $y$ out of $y (2 y^{2}) + y (- 13 y)$ .

$y (2 y^{2} - 13 y) + y \cdot - 7 = 0$

Factor $y$ out of $y (2 y^{2} - 13 y) + y \cdot - 7$ .

$y (2 y^{2} - 13 y - 7) = 0$

Step 2

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 7 = - 14$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 y$ .

$y (2 y^{2} - 13 y - 7) = 0$

Rewrite $- 13$ as $1$ plus $- 14$

$y (2 y^{2} + (1 - 14) y - 7) = 0$

Apply the distributive property.

$y (2 y^{2} + 1 y - 14 y - 7) = 0$

Multiply $y$ by $1$ .

$y (2 y^{2} + y - 14 y - 7) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y ((2 y^{2} + y) - 14 y - 7) = 0$

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

$y (y (2 y + 1) - 7 (2 y + 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 y + 1$ .

$y ((2 y + 1) (y - 7)) = 0$

Remove unnecessary parentheses.

$y (2 y + 1) (y - 7) = 0$

Step 3

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

$y = 0$

$2 y + 1 = 0$

$y - 7 = 0$

Step 4

Set $y$ equal to $0$ .

$y = 0$

Step 5

Set $2 y + 1$ equal to $0$ and solve for $y$ .

Set $2 y + 1$ equal to $0$ .

$2 y + 1 = 0$

Solve $2 y + 1 = 0$ for $y$ .

Subtract $1$ from both sides of the equation.

$2 y = - 1$

Divide each term in $2 y = - 1$ by $2$ and simplify.

Divide each term in $2 y = - 1$ by $2$ .

$\frac{2 y}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 1}{2}$

Divide $y$ by $1$ .

$y = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{1}{2}$

Step 6

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

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

$y - 7 = 0$

Add $7$ to both sides of the equation.

$y = 7$

Step 7

The final solution is all the values that make $y (2 y + 1) (y - 7) = 0$ true.

$y = 0, - \frac{1}{2}, 7$

Solve by Factoring 2y^3-13y^2-7y=0

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