Algebra Examples

$8 {(x + y)}^{3} - y^{3}$

Step 1

Rewrite $8 {(x + y)}^{3}$ as ${(2 (x + y))}^{3}$ .

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

Step 2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 2 (x + y)$ and $b = y$ .

$(2 (x + y) - y) ({(2 (x + y))}^{2} + 2 (x + y) y + y^{2})$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

$(2 x + 2 y - y) ({(2 (x + y))}^{2} + 2 (x + y) y + y^{2})$

Step 3.2

Subtract $y$ from $2 y$ .

$(2 x + y) ({(2 (x + y))}^{2} + 2 (x + y) y + y^{2})$

Step 3.3

Apply the distributive property.

$(2 x + y) ({(2 x + 2 y)}^{2} + 2 (x + y) y + y^{2})$

Step 3.4

Rewrite ${(2 x + 2 y)}^{2}$ as $(2 x + 2 y) (2 x + 2 y)$ .

$(2 x + y) ((2 x + 2 y) (2 x + 2 y) + 2 (x + y) y + y^{2})$

Step 3.5

Expand $(2 x + 2 y) (2 x + 2 y)$ using the FOIL Method.

Tap for more steps...

Step 3.5.1

Apply the distributive property.

$(2 x + y) (2 x (2 x + 2 y) + 2 y (2 x + 2 y) + 2 (x + y) y + y^{2})$

Step 3.5.2

Apply the distributive property.

$(2 x + y) (2 x (2 x) + 2 x (2 y) + 2 y (2 x + 2 y) + 2 (x + y) y + y^{2})$

Step 3.5.3

Apply the distributive property.

$(2 x + y) (2 x (2 x) + 2 x (2 y) + 2 y (2 x) + 2 y (2 y) + 2 (x + y) y + y^{2})$

Step 3.6

Simplify and combine like terms.

Tap for more steps...

Step 3.6.1

Simplify each term.

Tap for more steps...

Step 3.6.1.1

Rewrite using the commutative property of multiplication.

$(2 x + y) (2 \cdot 2 x \cdot x + 2 x (2 y) + 2 y (2 x) + 2 y (2 y) + 2 (x + y) y + y^{2})$

Step 3.6.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.6.1.2.1

Move $x$ .

$(2 x + y) (2 \cdot 2 (x \cdot x) + 2 x (2 y) + 2 y (2 x) + 2 y (2 y) + 2 (x + y) y + y^{2})$

Step 3.6.1.2.2

Multiply $x$ by $x$ .

$(2 x + y) (2 \cdot 2 x^{2} + 2 x (2 y) + 2 y (2 x) + 2 y (2 y) + 2 (x + y) y + y^{2})$

Step 3.6.1.3

Multiply $2$ by $2$ .

$(2 x + y) (4 x^{2} + 2 x (2 y) + 2 y (2 x) + 2 y (2 y) + 2 (x + y) y + y^{2})$

Step 3.6.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.6.1.5

Multiply $2$ by $2$ .

$(2 x + y) (4 x^{2} + 4 x y + 2 y (2 x) + 2 y (2 y) + 2 (x + y) y + y^{2})$

Step 3.6.1.6

Rewrite using the commutative property of multiplication.

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

Step 3.6.1.7

Multiply $2$ by $2$ .

$(2 x + y) (4 x^{2} + 4 x y + 4 y x + 2 y (2 y) + 2 (x + y) y + y^{2})$

Step 3.6.1.8

Rewrite using the commutative property of multiplication.

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

Step 3.6.1.9

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 3.6.1.9.1

Move $y$ .

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

Step 3.6.1.9.2

Multiply $y$ by $y$ .

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

Step 3.6.1.10

Multiply $2$ by $2$ .

$(2 x + y) (4 x^{2} + 4 x y + 4 y x + 4 y^{2} + 2 (x + y) y + y^{2})$

Step 3.6.2

Add $4 x y$ and $4 y x$ .

Tap for more steps...

Step 3.6.2.1

Move $y$ .

$(2 x + y) (4 x^{2} + 4 x y + 4 x y + 4 y^{2} + 2 (x + y) y + y^{2})$

Step 3.6.2.2

Add $4 x y$ and $4 x y$ .

$(2 x + y) (4 x^{2} + 8 x y + 4 y^{2} + 2 (x + y) y + y^{2})$

Step 3.7

Apply the distributive property.

$(2 x + y) (4 x^{2} + 8 x y + 4 y^{2} + (2 x + 2 y) y + y^{2})$

Step 3.8

Apply the distributive property.

$(2 x + y) (4 x^{2} + 8 x y + 4 y^{2} + 2 x y + 2 y \cdot y + y^{2})$

Step 3.9

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 3.9.1

Move $y$ .

$(2 x + y) (4 x^{2} + 8 x y + 4 y^{2} + 2 x y + 2 (y \cdot y) + y^{2})$

Step 3.9.2

Multiply $y$ by $y$ .

$(2 x + y) (4 x^{2} + 8 x y + 4 y^{2} + 2 x y + 2 y^{2} + y^{2})$

Step 3.10

Add $8 x y$ and $2 x y$ .

$(2 x + y) (4 x^{2} + 4 y^{2} + 10 x y + 2 y^{2} + y^{2})$

Step 3.11

Add $4 y^{2}$ and $2 y^{2}$ .

$(2 x + y) (4 x^{2} + 6 y^{2} + 10 x y + y^{2})$

Step 3.12

Add $6 y^{2}$ and $y^{2}$ .

$(2 x + y) (4 x^{2} + 7 y^{2} + 10 x y)$