Algebra Examples

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

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

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

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

Step 3

Simplify each term.

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Step 3.1

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

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Step 3.1.1

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.2

Add $1$ and $2$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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Step 3.3.1

Move $x$ .

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

Step 3.3.2

Multiply $x$ by $x$ .

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

Step 3.4

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

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Step 3.4.1

Move $y$ .

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

Step 3.4.2

Multiply $y$ by $y$ .

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

Step 3.5

Rewrite using the commutative property of multiplication.

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

Step 3.6

Multiply $- 1$ by $2$ .

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

Step 3.7

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

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Step 3.7.1

Move $y^{2}$ .

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

Step 3.7.2

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

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Step 3.7.2.1

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

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

Step 3.7.2.2

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

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

Step 3.7.3

Add $2$ and $1$ .

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

Step 4

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

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Step 4.1

Move $y$ .

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

Step 4.2

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

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

Step 5

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

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

Step 6

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

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Step 6.1

Move $y^{2}$ .

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

Step 6.2

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

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