Algebra Examples

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

Step 1

Apply the distributive property.

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

Step 2

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

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Step 2.1

Move $x$ .

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

Step 2.2

Multiply $x$ by $x$ .

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

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

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

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Step 4.1

Move $y$ .

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

Step 4.2

Multiply $y$ by $y$ .

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

Step 5

Rewrite $x^{2} y - x y^{2} + x^{3} - y^{3}$ in a factored form.

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Step 5.1

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

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Step 5.1.1

Factor $x y$ out of $x^{2} y$ .

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

Step 5.1.2

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

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

Step 5.1.3

Factor $x y$ out of $x y (x) + x y (- y)$ .

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

Factor $x - y$ out of $x y (x - y)$ .

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

Step 5.3.2

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

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

Step 5.4

Add $x y$ and $x y$ .

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

Step 5.5

Factor using the perfect square rule.

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Step 5.5.1

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

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

Step 5.5.2

Rewrite the polynomial.

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

Step 5.5.3

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

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