Algebra Examples

$1 + {(x + y)}^{3}$

Step 1

Rewrite $1$ as $1^{3}$ .

$1^{3} + {(x + y)}^{3}$

Step 2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = x + y$ .

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

Step 3

Simplify.

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Step 3.1

One to any power is one.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Rewrite $- 1 x$ as $- x$ .

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

Step 3.4

Rewrite $- 1 y$ as $- y$ .

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

Step 3.5

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

$(1 + x + y) (1 - x - y + (x + y) (x + y))$

Step 3.6

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

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Step 3.6.1

Apply the distributive property.

$(1 + x + y) (1 - x - y + x (x + y) + y (x + y))$

Step 3.6.2

Apply the distributive property.

$(1 + x + y) (1 - x - y + x \cdot x + x y + y (x + y))$

Step 3.6.3

Apply the distributive property.

$(1 + x + y) (1 - x - y + x \cdot x + x y + y x + y \cdot y)$

Step 3.7

Simplify and combine like terms.

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Step 3.7.1

Simplify each term.

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Step 3.7.1.1

Multiply $x$ by $x$ .

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

Step 3.7.1.2

Multiply $y$ by $y$ .

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

Step 3.7.2

Add $x y$ and $y x$ .

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Step 3.7.2.1

Reorder $y$ and $x$ .

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

Step 3.7.2.2

Add $x y$ and $x y$ .

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