Simplify (x-y)(x^2+2xy-y^2)

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

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.

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

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

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

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

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

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

Add $1$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $- 1$ by $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $y^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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

Step 3

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

Move $y$ .

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

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

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

Step 4

Multiply $x^{2}$ by $1$ .

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

Step 5

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

Move $y^{2}$ .

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

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

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

Simplify (x-y)(x^2+2xy-y^2)

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