Algebra Examples

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

Step 1

Factor $3 x$ out of $3 x^{2} {(x - 2 y)}^{2} + 9 x y^{2} (2 y - x)$ .

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Step 1.1

Factor $3 x$ out of $3 x^{2} {(x - 2 y)}^{2}$ .

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

Step 1.2

Factor $3 x$ out of $9 x y^{2} (2 y - x)$ .

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

Step 1.3

Factor $3 x$ out of $3 x (x {(x - 2 y)}^{2}) + 3 x (3 y^{2} (2 y - x))$ .

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

Step 2

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

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

Step 3

Expand $(x - 2 y) (x - 2 y)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $x$ by $x$ .

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

Step 4.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.1.4

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

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Step 4.1.4.1

Move $y$ .

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

Step 4.1.4.2

Multiply $y$ by $y$ .

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

Step 4.1.5

Multiply $- 2$ by $- 2$ .

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

Step 4.2

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

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Step 4.2.1

Move $y$ .

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

Step 4.2.2

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

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

Step 5

Apply the distributive property.

$3 x (x \cdot x^{2} + x (- 4 x y) + x (4 y^{2}) + 3 y^{2} (2 y - x))$

Step 6

Simplify.

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Step 6.1

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

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Step 6.1.1

Multiply $x$ by $x^{2}$ .

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Step 6.1.1.1

Raise $x$ to the power of $1$ .

$3 x (x^{1} x^{2} + x (- 4 x y) + x (4 y^{2}) + 3 y^{2} (2 y - x))$

Step 6.1.1.2

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

$3 x (x^{1 + 2} + x (- 4 x y) + x (4 y^{2}) + 3 y^{2} (2 y - x))$

Step 6.1.2

Add $1$ and $2$ .

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

Step 6.2

Rewrite using the commutative property of multiplication.

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

Step 6.3

Rewrite using the commutative property of multiplication.

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

Step 7

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

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Step 7.1

Move $x$ .

$3 x (x^{3} - 4 (x \cdot x) y + 4 x y^{2} + 3 y^{2} (2 y - x))$

Step 7.2

Multiply $x$ by $x$ .

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

Step 8

Apply the distributive property.

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

Step 9

Rewrite using the commutative property of multiplication.

$3 x (x^{3} - 4 x^{2} y + 4 x y^{2} + 3 \cdot 2 y^{2} y + 3 y^{2} (- x))$

Step 10

Rewrite using the commutative property of multiplication.

$3 x (x^{3} - 4 x^{2} y + 4 x y^{2} + 3 \cdot 2 y^{2} y + 3 \cdot - 1 y^{2} x)$

Step 11

Simplify each term.

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Step 11.1

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

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Step 11.1.1

Move $y$ .

$3 x (x^{3} - 4 x^{2} y + 4 x y^{2} + 3 \cdot 2 (y \cdot y^{2}) + 3 \cdot - 1 y^{2} x)$

Step 11.1.2

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

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Step 11.1.2.1

Raise $y$ to the power of $1$ .

$3 x (x^{3} - 4 x^{2} y + 4 x y^{2} + 3 \cdot 2 (y^{1} y^{2}) + 3 \cdot - 1 y^{2} x)$

Step 11.1.2.2

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

$3 x (x^{3} - 4 x^{2} y + 4 x y^{2} + 3 \cdot 2 y^{1 + 2} + 3 \cdot - 1 y^{2} x)$

Step 11.1.3

Add $1$ and $2$ .

$3 x (x^{3} - 4 x^{2} y + 4 x y^{2} + 3 \cdot 2 y^{3} + 3 \cdot - 1 y^{2} x)$

Step 11.2

Multiply $3$ by $2$ .

$3 x (x^{3} - 4 x^{2} y + 4 x y^{2} + 6 y^{3} + 3 \cdot - 1 y^{2} x)$

Step 11.3

Multiply $3$ by $- 1$ .

$3 x (x^{3} - 4 x^{2} y + 4 x y^{2} + 6 y^{3} - 3 y^{2} x)$

Step 12

Subtract $3 y^{2} x$ from $4 x y^{2}$ .

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Step 12.1

Move $y^{2}$ .

$3 x (x^{3} - 4 x^{2} y + 6 y^{3} + 4 x y^{2} - 3 x y^{2})$

Step 12.2

Subtract $3 x y^{2}$ from $4 x y^{2}$ .

$3 x (x^{3} - 4 x^{2} y + 6 y^{3} + 1 x y^{2})$

Step 13

Multiply $x$ by $1$ .

$3 x (x^{3} - 4 x^{2} y + 6 y^{3} + x y^{2})$