Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Factor $x (2 x - y)$ out of $x (2 x - y) (x^{2}) + x (2 x - y) (- 1 y (2 x - y))$ .

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

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

Rewrite using the commutative property of multiplication.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Simplify each term.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

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

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Step 6.2.1

Move $y$ .

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

Step 6.2.2

Multiply $y$ by $y$ .

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

Step 6.3

Multiply $- 1$ by $- 1$ .

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

Step 6.4

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

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

Step 7

Factor using the perfect square rule.

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Step 7.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 y x = 2 \cdot x \cdot y$

Step 7.2

Rewrite the polynomial.

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

Step 7.3

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = y$ .

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