Algebra Examples

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

Step 1

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

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

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 $1$ .

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

Step 2.1.1.1.2

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

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

Step 2.1.1.2

Add $2$ and $1$ .

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

Step 2.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.1.3

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

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Step 2.1.3.1

Move $x$ .

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

Step 2.1.3.2

Multiply $x$ by $x$ .

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

Step 2.1.4

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

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Step 2.1.4.1

Move $y$ .

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

Step 2.1.4.2

Multiply $y$ by $y$ .

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

Step 2.1.5

Multiply $- 2$ by $2$ .

$x^{3} - 2 x^{2} y + 2 x^{2} y - 4 x y^{2} + y^{2} x + y^{2} (- 2 y)$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$x^{3} - 2 x^{2} y + 2 x^{2} y - 4 x y^{2} + y^{2} x - 2 y^{2} y$

Step 2.1.7

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

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Step 2.1.7.1

Move $y$ .

$x^{3} - 2 x^{2} y + 2 x^{2} y - 4 x y^{2} + y^{2} x - 2 (y \cdot y^{2})$

Step 2.1.7.2

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

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Step 2.1.7.2.1

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

$x^{3} - 2 x^{2} y + 2 x^{2} y - 4 x y^{2} + y^{2} x - 2 (y^{1} y^{2})$

Step 2.1.7.2.2

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

$x^{3} - 2 x^{2} y + 2 x^{2} y - 4 x y^{2} + y^{2} x - 2 y^{1 + 2}$

Step 2.1.7.3

Add $1$ and $2$ .

$x^{3} - 2 x^{2} y + 2 x^{2} y - 4 x y^{2} + y^{2} x - 2 y^{3}$

Step 2.2

Combine the opposite terms in $x^{3} - 2 x^{2} y + 2 x^{2} y - 4 x y^{2} + y^{2} x - 2 y^{3}$ .

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Step 2.2.1

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

$x^{3} + 0 - 4 x y^{2} + y^{2} x - 2 y^{3}$

Step 2.2.2

Add $x^{3}$ and $0$ .

$x^{3} - 4 x y^{2} + y^{2} x - 2 y^{3}$

Step 3

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

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Step 3.1

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

$x^{3} - 4 x y^{2} + x y^{2} - 2 y^{3}$

Step 3.2

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

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