Pre-Algebra Examples

$(8 x^{3} + 2 x^{2} y - 8 x y^{2} - 2 y^{3}) \div (4 x^{2} - 3 x y - y^{2})$

Step 1

Rewrite the division as a fraction.

$\frac{8 x^{3} + 2 x^{2} y - 8 x y^{2} - 2 y^{3}}{4 x^{2} - 3 x y - y^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2$ out of $8 x^{3} + 2 x^{2} y - 8 x y^{2} - 2 y^{3}$ .

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Step 2.1.1

Factor $2$ out of $8 x^{3}$ .

$\frac{2 (4 x^{3}) + 2 x^{2} y - 8 x y^{2} - 2 y^{3}}{4 x^{2} - 3 x y - y^{2}}$

Step 2.1.2

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

$\frac{2 (4 x^{3}) + 2 (x^{2} y) - 8 x y^{2} - 2 y^{3}}{4 x^{2} - 3 x y - y^{2}}$

Step 2.1.3

Factor $2$ out of $- 8 x y^{2}$ .

$\frac{2 (4 x^{3}) + 2 (x^{2} y) + 2 (- 4 x y^{2}) - 2 y^{3}}{4 x^{2} - 3 x y - y^{2}}$

Step 2.1.4

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

$\frac{2 (4 x^{3}) + 2 (x^{2} y) + 2 (- 4 x y^{2}) + 2 (- y^{3})}{4 x^{2} - 3 x y - y^{2}}$

Step 2.1.5

Factor $2$ out of $2 (4 x^{3}) + 2 (x^{2} y)$ .

$\frac{2 (4 x^{3} + x^{2} y) + 2 (- 4 x y^{2}) + 2 (- y^{3})}{4 x^{2} - 3 x y - y^{2}}$

Step 2.1.6

Factor $2$ out of $2 (4 x^{3} + x^{2} y) + 2 (- 4 x y^{2})$ .

$\frac{2 (4 x^{3} + x^{2} y - 4 x y^{2}) + 2 (- y^{3})}{4 x^{2} - 3 x y - y^{2}}$

Step 2.1.7

Factor $2$ out of $2 (4 x^{3} + x^{2} y - 4 x y^{2}) + 2 (- y^{3})$ .

$\frac{2 (4 x^{3} + x^{2} y - 4 x y^{2} - y^{3})}{4 x^{2} - 3 x y - y^{2}}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{2 ((4 x^{3} + x^{2} y) - 4 x y^{2} - y^{3})}{4 x^{2} - 3 x y - y^{2}}$

Step 2.2.2

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

$\frac{2 (x^{2} (4 x + y) - y^{2} (4 x + y))}{4 x^{2} - 3 x y - y^{2}}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $4 x + y$ .

$\frac{2 ((4 x + y) (x^{2} - y^{2}))}{4 x^{2} - 3 x y - y^{2}}$

Step 2.4

Factor.

$\frac{2 (4 x + y) (x + y) (x - y)}{4 x^{2} - 3 x y - y^{2}}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 4 \cdot - 1 = - 4$ and whose sum is $b = - 3$ .

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Step 3.1.1

Reorder terms.

$\frac{2 (4 x + y) (x + y) (x - y)}{4 x^{2} - y^{2} - 3 x y}$

Step 3.1.2

Reorder $- y^{2}$ and $- 3 x y$ .

$\frac{2 (4 x + y) (x + y) (x - y)}{4 x^{2} - 3 x y - y^{2}}$

Step 3.1.3

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

$\frac{2 (4 x + y) (x + y) (x - y)}{4 x^{2} - 3 (x y) - y^{2}}$

Step 3.1.4

Rewrite $- 3$ as $1$ plus $- 4$

$\frac{2 (4 x + y) (x + y) (x - y)}{4 x^{2} + (1 - 4) (x y) - y^{2}}$

Step 3.1.5

Apply the distributive property.

$\frac{2 (4 x + y) (x + y) (x - y)}{4 x^{2} + 1 (x y) - 4 (x y) - y^{2}}$

Step 3.1.6

Multiply $x y$ by $1$ .

$\frac{2 (4 x + y) (x + y) (x - y)}{4 x^{2} + x y - 4 (x y) - y^{2}}$

Step 3.1.7

Move parentheses.

$\frac{2 (4 x + y) (x + y) (x - y)}{4 x^{2} + x y - 4 x y - y^{2}}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{2 (4 x + y) (x + y) (x - y)}{(4 x^{2} + x y) - 4 x y - y^{2}}$

Step 3.2.2

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

$\frac{2 (4 x + y) (x + y) (x - y)}{x (4 x + y) - y (4 x + y)}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $4 x + y$ .

$\frac{2 (4 x + y) (x + y) (x - y)}{(4 x + y) (x - y)}$

Step 4

Cancel the common factor of $4 x + y$ .

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Step 4.1

Cancel the common factor.

$\frac{2 (4 x + y) (x + y) (x - y)}{(4 x + y) (x - y)}$

Step 4.2

Rewrite the expression.

$\frac{(2 (x + y)) (x - y)}{x - y}$

Step 5

Cancel the common factor of $x - y$ .

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Step 5.1

Cancel the common factor.

$\frac{2 (x + y) (x - y)}{x - y}$

Step 5.2

Divide $2 (x + y)$ by $1$ .

$2 (x + y)$

Step 6

Apply the distributive property.

$2 x + 2 y$