Algebra Examples

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

Step 1

Rewrite ${(x + y)}^{2}$ as $(x + y) (x + y)$ .

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

Step 2

Expand $(x + y) (x + y)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Multiply $y$ by $y$ .

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

Step 3.2

Add $x y$ and $y x$ .

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Step 3.2.1

Reorder $y$ and $x$ .

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

Step 3.2.2

Add $x y$ and $x y$ .

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

Step 4

Rewrite ${(x - y)}^{2}$ as $(x - y) (x - y)$ .

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

Step 5

Expand $(x - y) (x - y)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

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

Step 6.1.2

Rewrite using the commutative property of multiplication.

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

Step 6.1.3

Rewrite using the commutative property of multiplication.

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

Step 6.1.4

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

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Step 6.1.4.1

Move $y$ .

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

Step 6.1.4.2

Multiply $y$ by $y$ .

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

Step 6.1.5

Multiply $- 1$ by $- 1$ .

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

Step 6.1.6

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

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

Step 6.2

Subtract $y x$ from $- x y$ .

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Step 6.2.1

Move $y$ .

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

Step 6.2.2

Subtract $x y$ from $- x y$ .

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

Step 7

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

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

Step 8

Simplify terms.

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Step 8.1

Combine the opposite terms in $x^{2} x^{2} + x^{2} (- 2 x y) + x^{2} y^{2} + 2 x y x^{2} + 2 x y (- 2 x y) + 2 x y \cdot y^{2} + y^{2} x^{2} + y^{2} (- 2 x y) + y^{2} y^{2}$ .

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Step 8.1.1

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

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

Step 8.1.2

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

$x^{2} x^{2} + x^{2} y^{2} + 0 + 2 x y (- 2 x y) + 2 x y \cdot y^{2} + y^{2} x^{2} + y^{2} (- 2 x y) + y^{2} y^{2}$

Step 8.1.3

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

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

Step 8.1.4

Reorder the factors in the terms $2 x y \cdot y^{2}$ and $y^{2} (- 2 x y)$ .

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

Step 8.1.5

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

$x^{2} x^{2} + x^{2} y^{2} + 2 x y (- 2 x y) + y^{2} x^{2} + 0 + y^{2} y^{2}$

Step 8.1.6

Add $x^{2} x^{2} + x^{2} y^{2} + 2 x y (- 2 x y) + y^{2} x^{2}$ and $0$ .

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

Step 8.2

Simplify each term.

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Step 8.2.1

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

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Step 8.2.1.1

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

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

Step 8.2.1.2

Add $2$ and $2$ .

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

Step 8.2.2

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

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Step 8.2.2.1

Move $x$ .

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

Step 8.2.2.2

Multiply $x$ by $x$ .

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

Step 8.2.3

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

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Step 8.2.3.1

Move $y$ .

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

Step 8.2.3.2

Multiply $y$ by $y$ .

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

Step 8.2.4

Multiply $- 2$ by $2$ .

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

Step 8.2.5

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

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Step 8.2.5.1

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

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

Step 8.2.5.2

Add $2$ and $2$ .

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

Step 8.3

Subtract $4 x^{2} y^{2}$ from $x^{2} y^{2}$ .

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

Step 9

Add $- 3 x^{2} y^{2}$ and $y^{2} x^{2}$ .

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Step 9.1

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

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

Step 9.2

Add $- 3 x^{2} y^{2}$ and $x^{2} y^{2}$ .

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