Algebra Examples

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

Step 1

Rewrite ${(x + y)}^{4}$ as ${({(x + y)}^{2})}^{2}$ .

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

Step 2

Rewrite $1$ as $1^{2}$ .

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

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x + y)}^{2}$ and $b = 1$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $x$ by $x$ .

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

Step 4.3.1.2

Multiply $y$ by $y$ .

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

Step 4.3.2

Add $x y$ and $y x$ .

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Step 4.3.2.1

Reorder $y$ and $x$ .

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

Step 4.3.2.2

Add $x y$ and $x y$ .

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

Step 4.4

Rewrite $1$ as $1^{2}$ .

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

Step 4.5

Factor.

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Step 4.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + y$ and $b = 1$ .

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

Step 4.5.2

Remove unnecessary parentheses.

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