Algebra Examples

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

Step 1

Rewrite the division as a fraction.

$\frac{x^{3} + y^{3}}{x - y}$

Step 2

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

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

Step 3

Expand $(x + y) (x^{2} - x y + y^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4

Simplify each term.

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Step 4.1

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

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Step 4.1.1

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

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.2

Add $1$ and $2$ .

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

Step 4.2

Rewrite using the commutative property of multiplication.

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

Step 4.3

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

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Step 4.3.1

Move $x$ .

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

Step 4.3.2

Multiply $x$ by $x$ .

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

Step 4.4

Rewrite using the commutative property of multiplication.

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

Step 4.5

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

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Step 4.5.1

Move $y$ .

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

Step 4.5.2

Multiply $y$ by $y$ .

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

Step 4.6

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

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Step 4.6.1

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

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Step 4.6.1.1

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

$\frac{x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{1} y^{2}}{x - y}$

Step 4.6.1.2

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

$\frac{x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{1 + 2}}{x - y}$

Step 4.6.2

Add $1$ and $2$ .

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

Step 5

Combine the opposite terms in $x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{3}$ .

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Step 5.1

Reorder the factors in the terms $- x^{2} y$ and $y x^{2}$ .

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

Step 5.2

Add $- x^{2} y$ and $x^{2} y$ .

$\frac{x^{3} + x y^{2} + 0 - y^{2} x + y^{3}}{x - y}$

Step 5.3

Add $x^{3} + x y^{2}$ and $0$ .

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

Step 5.4

Reorder the factors in the terms $x y^{2}$ and $- y^{2} x$ .

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

Step 5.5

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

$\frac{x^{3} + 0 + y^{3}}{x - y}$

Step 5.6

Add $x^{3}$ and $0$ .

$\frac{x^{3} + y^{3}}{x - y}$

Step 6

Split the fraction $\frac{x^{3} + y^{3}}{x - y}$ into two fractions.

$\frac{x^{3}}{x - y} + \frac{y^{3}}{x - y}$