Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

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

Step 2.1.1.1.2

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

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

Step 2.1.1.2

Add $2$ and $1$ .

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

Step 2.1.2

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

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Step 2.1.2.1

Move $x$ .

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

Step 2.1.2.2

Multiply $x$ by $x$ .

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

Step 2.1.3

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

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Step 2.1.3.1

Move $y$ .

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

Step 2.1.3.2

Multiply $y$ by $y$ .

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

Step 2.1.4

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

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Step 2.1.4.1

Move $y$ .

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

Step 2.1.4.2

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

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Step 2.1.4.2.1

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

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

Step 2.1.4.2.2

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

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

Step 2.1.4.3

Add $1$ and $2$ .

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

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

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Step 2.2.1.1

Reorder the factors in the terms $x y^{2}$ and $- y^{2} x$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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