Algebra Examples

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

Step 1

Expand $(x^{3} - x^{2} - x) (x^{3} + x^{2} + x)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Reorder the factors in the terms $x^{3} x^{2}$ and $- x^{2} x^{3}$ .

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

Step 2.1.2

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

$x^{3} x^{3} + x^{3} x + 0 - x^{2} x^{2} - x^{2} x - x \cdot x^{3} - x \cdot x^{2} - x \cdot x$

Step 2.1.3

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

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

Step 2.1.4

Reorder the factors in the terms $x^{3} x$ and $- x \cdot x^{3}$ .

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

Step 2.1.5

Subtract $x^{3} x$ from $x^{3} x$ .

$x^{3} x^{3} - x^{2} x^{2} - x^{2} x + 0 - x \cdot x^{2} - x \cdot x$

Step 2.1.6

Add $x^{3} x^{3} - x^{2} x^{2} - x^{2} x$ and $0$ .

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

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 2.2.1.1

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

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

Step 2.2.1.2

Add $3$ and $3$ .

$x^{6} - x^{2} x^{2} - x^{2} x - x \cdot x^{2} - x \cdot x$

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

Move $x^{2}$ .

$x^{6} - (x^{2} x^{2}) - x^{2} x - x \cdot x^{2} - x \cdot x$

Step 2.2.2.2

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

$x^{6} - x^{2 + 2} - x^{2} x - x \cdot x^{2} - x \cdot x$

Step 2.2.2.3

Add $2$ and $2$ .

$x^{6} - x^{4} - x^{2} x - x \cdot x^{2} - x \cdot x$

Step 2.2.3

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

Tap for more steps...

Step 2.2.3.1

Move $x$ .

$x^{6} - x^{4} - (x \cdot x^{2}) - x \cdot x^{2} - x \cdot x$

Step 2.2.3.2

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

Tap for more steps...

Step 2.2.3.2.1

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

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

Step 2.2.3.2.2

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

$x^{6} - x^{4} - x^{1 + 2} - x \cdot x^{2} - x \cdot x$

Step 2.2.3.3

Add $1$ and $2$ .

$x^{6} - x^{4} - x^{3} - x \cdot x^{2} - x \cdot x$

Step 2.2.4

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

Tap for more steps...

Step 2.2.4.1

Move $x^{2}$ .

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

Step 2.2.4.2

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

Tap for more steps...

Step 2.2.4.2.1

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

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

Step 2.2.4.2.2

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

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

Step 2.2.4.3

Add $2$ and $1$ .

$x^{6} - x^{4} - x^{3} - x^{3} - x \cdot x$

Step 2.2.5

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

Tap for more steps...

Step 2.2.5.1

Move $x$ .

$x^{6} - x^{4} - x^{3} - x^{3} - (x \cdot x)$

Step 2.2.5.2

Multiply $x$ by $x$ .

$x^{6} - x^{4} - x^{3} - x^{3} - x^{2}$

Step 2.3

Subtract $x^{3}$ from $- x^{3}$ .

$x^{6} - x^{4} - 2 x^{3} - x^{2}$