Algebra Examples

$27 x^{3} - {(x - y)}^{3}$

Step 1

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

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

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 $a = 3 x$ and $b = x - y$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- - y$ .

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2

Multiply $y$ by $1$ .

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

Step 3.3

Subtract $x$ from $3 x$ .

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

Step 3.4

Apply the product rule to $3 x$ .

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

Step 3.5

Raise $3$ to the power of $2$ .

$(2 x + y) (9 x^{2} + 3 x (x - y) + {(x - y)}^{2})$

Step 3.6

Apply the distributive property.

$(2 x + y) (9 x^{2} + 3 x \cdot x + 3 x (- y) + {(x - y)}^{2})$

Step 3.7

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

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Step 3.7.1

Move $x$ .

$(2 x + y) (9 x^{2} + 3 (x \cdot x) + 3 x (- y) + {(x - y)}^{2})$

Step 3.7.2

Multiply $x$ by $x$ .

$(2 x + y) (9 x^{2} + 3 x^{2} + 3 x (- y) + {(x - y)}^{2})$

Step 3.8

Rewrite using the commutative property of multiplication.

$(2 x + y) (9 x^{2} + 3 x^{2} + 3 \cdot - 1 x y + {(x - y)}^{2})$

Step 3.9

Multiply $3$ by $- 1$ .

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + {(x - y)}^{2})$

Step 3.10

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

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + (x - y) (x - y))$

Step 3.11

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

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Step 3.11.1

Apply the distributive property.

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x (x - y) - y (x - y))$

Step 3.11.2

Apply the distributive property.

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x \cdot x + x (- y) - y (x - y))$

Step 3.11.3

Apply the distributive property.

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x \cdot x + x (- y) - y x - y (- y))$

Step 3.12

Simplify and combine like terms.

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Step 3.12.1

Simplify each term.

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Step 3.12.1.1

Multiply $x$ by $x$ .

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} + x (- y) - y x - y (- y))$

Step 3.12.1.2

Rewrite using the commutative property of multiplication.

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} - x y - y x - y (- y))$

Step 3.12.1.3

Rewrite using the commutative property of multiplication.

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} - x y - y x - 1 \cdot - 1 y \cdot y)$

Step 3.12.1.4

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

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Step 3.12.1.4.1

Move $y$ .

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} - x y - y x - 1 \cdot - 1 (y \cdot y))$

Step 3.12.1.4.2

Multiply $y$ by $y$ .

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} - x y - y x - 1 \cdot - 1 y^{2})$

Step 3.12.1.5

Multiply $- 1$ by $- 1$ .

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} - x y - y x + 1 y^{2})$

Step 3.12.1.6

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

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} - x y - y x + y^{2})$

Step 3.12.2

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

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Step 3.12.2.1

Move $y$ .

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} - x y - 1 x y + y^{2})$

Step 3.12.2.2

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

$(2 x + y) (9 x^{2} + 3 x^{2} - 3 x y + x^{2} - 2 x y + y^{2})$

Step 3.13

Add $9 x^{2}$ and $3 x^{2}$ .

$(2 x + y) (12 x^{2} - 3 x y + x^{2} - 2 x y + y^{2})$

Step 3.14

Add $12 x^{2}$ and $x^{2}$ .

$(2 x + y) (13 x^{2} - 3 x y - 2 x y + y^{2})$

Step 3.15

Subtract $2 x y$ from $- 3 x y$ .

$(2 x + y) (13 x^{2} - 5 x y + y^{2})$