Algebra Examples

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

Step 1

Expand $(x + y) (x^{2} + 2 x y - y^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify each term.

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Step 2.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.1.1

Multiply $x$ by $x^{2}$ .

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Step 2.1.1.1

Raise $x$ to the power of $1$ .

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

Step 2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.2

Add $1$ and $2$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

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

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Step 2.3.1

Move $x$ .

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

Step 2.3.2

Multiply $x$ by $x$ .

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

Step 2.4

Rewrite using the commutative property of multiplication.

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

Step 2.5

Rewrite using the commutative property of multiplication.

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

Step 2.6

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

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Step 2.6.1

Move $y$ .

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

Step 2.6.2

Multiply $y$ by $y$ .

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

Step 2.7

Rewrite using the commutative property of multiplication.

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

Step 2.8

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 2.8.1

Move $y^{2}$ .

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

Step 2.8.2

Multiply $y^{2}$ by $y$ .

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Step 2.8.2.1

Raise $y$ to the power of $1$ .

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

Step 2.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.8.3

Add $2$ and $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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Step 4.1

Move $y^{2}$ .

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

Step 4.2

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

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

Step 5

Multiply $x$ by $1$ .

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