Algebra Examples

${(x + 1)}^{3} - 64 y^{3}$

Step 1

Rewrite $64 y^{3}$ as ${(4 y)}^{3}$ .

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

Step 2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x + 1$ and $b = 4 y$ .

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

Step 3

Simplify.

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Step 3.1

Multiply $4$ by $- 1$ .

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

Step 3.2

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Step 3.3

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Apply the distributive property.

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

Step 3.4

Simplify and combine like terms.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $x$ by $x$ .

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

Step 3.4.1.2

Multiply $x$ by $1$ .

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

Step 3.4.1.3

Multiply $x$ by $1$ .

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

Step 3.4.1.4

Multiply $1$ by $1$ .

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

Step 3.4.2

Add $x$ and $x$ .

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

Step 3.5

Apply the distributive property.

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

Step 3.6

Rewrite using the commutative property of multiplication.

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

Step 3.7

Multiply $4 y$ by $1$ .

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

Step 3.8

Apply the product rule to $4 y$ .

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

Step 3.9

Raise $4$ to the power of $2$ .

$(x + 1 - 4 y) (x^{2} + 2 x + 1 + 4 x y + 4 y + 16 y^{2})$