Algebra Examples

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

Step 1

Expand $(x y - x^{2} y) (2 x - 3 x y + y)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.2

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

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Step 2.1.2.1

Move $x$ .

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

Step 2.1.2.2

Multiply $x$ by $x$ .

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

Step 2.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.1.4

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

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Step 2.1.4.1

Move $x$ .

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

Step 2.1.4.2

Multiply $x$ by $x$ .

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

Step 2.1.5

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

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Step 2.1.5.1

Move $y$ .

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

Step 2.1.5.2

Multiply $y$ by $y$ .

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

Step 2.1.6

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

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Step 2.1.6.1

Move $y$ .

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

Step 2.1.6.2

Multiply $y$ by $y$ .

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

Step 2.1.7

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

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Step 2.1.7.1

Move $x$ .

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

Step 2.1.7.2

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

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Step 2.1.7.2.1

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

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

Step 2.1.7.2.2

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

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

Step 2.1.7.3

Add $1$ and $2$ .

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

Step 2.1.8

Multiply $2$ by $- 1$ .

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

Step 2.1.9

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

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Step 2.1.9.1

Move $x$ .

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

Step 2.1.9.2

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

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Step 2.1.9.2.1

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

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

Step 2.1.9.2.2

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

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

Step 2.1.9.3

Add $1$ and $2$ .

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

Step 2.1.10

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

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Step 2.1.10.1

Move $y$ .

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

Step 2.1.10.2

Multiply $y$ by $y$ .

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

Step 2.1.11

Multiply $- 3$ by $- 1$ .

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

Step 2.1.12

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

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Step 2.1.12.1

Move $y$ .

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

Step 2.1.12.2

Multiply $y$ by $y$ .

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

Step 2.2

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

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