Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Use the Binomial Theorem.

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

Step 1.3

Simplify each term.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3} - (x^{3} + 3 \cdot - 1 x^{2} y + 3 x {(- y)}^{2} + {(- y)}^{3})$

Step 1.3.2

Multiply $3$ by $- 1$ .

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

Step 1.3.3

Apply the product rule to $- y$ .

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

Step 1.3.4

Rewrite using the commutative property of multiplication.

$x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3} - (x^{3} - 3 x^{2} y + 3 \cdot {(- 1)}^{2} x y^{2} + {(- y)}^{3})$

Step 1.3.5

Raise $- 1$ to the power of $2$ .

$x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3} - (x^{3} - 3 x^{2} y + 3 \cdot 1 x y^{2} + {(- y)}^{3})$

Step 1.3.6

Multiply $3$ by $1$ .

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

Step 1.3.7

Apply the product rule to $- y$ .

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

Step 1.3.8

Raise $- 1$ to the power of $3$ .

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 3$ by $- 1$ .

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

Step 1.5.2

Multiply $3$ by $- 1$ .

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

Step 1.5.3

Multiply $- - y^{3}$ .

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Step 1.5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.5.3.2

Multiply $y^{3}$ by $1$ .

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

Step 1.6

Remove parentheses.

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

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3} - x^{3} + 3 x^{2} y - 3 x y^{2} + y^{3}$ .

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Step 2.1.1

Subtract $x^{3}$ from $x^{3}$ .

$3 x^{2} y + 3 x y^{2} + y^{3} + 0 + 3 x^{2} y - 3 x y^{2} + y^{3}$

Step 2.1.2

Add $3 x^{2} y + 3 x y^{2} + y^{3}$ and $0$ .

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

Step 2.1.3

Subtract $3 x y^{2}$ from $3 x y^{2}$ .

$3 x^{2} y + y^{3} + 3 x^{2} y + 0 + y^{3}$

Step 2.1.4

Add $3 x^{2} y + y^{3} + 3 x^{2} y$ and $0$ .

$3 x^{2} y + y^{3} + 3 x^{2} y + y^{3}$

Step 2.2

Add $3 x^{2} y$ and $3 x^{2} y$ .

$y^{3} + 6 x^{2} y + y^{3}$

Step 2.3

Add $y^{3}$ and $y^{3}$ .

$2 y^{3} + 6 x^{2} y$