Algebra Examples

x^{3} + 3 x + 3

$x^{3} + 3 x + 3$ ,

- x^{3} - 3

$- x^{3} - 3$

Step 1

Multiply the expressions.

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

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

Step 2

Expand

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

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

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

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

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

Step 3.1.2

Multiply

x^{3}

$x^{3}$ by

x^{3}

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

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Step 3.1.2.1

Move

x^{3}

$x^{3}$ .

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

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

Step 3.1.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.1.2.3

Add

3

$3$ and

3

$3$ .

- x^{6} + x^{3} \cdot - 3 + 3 x (- x^{3}) + 3 x \cdot - 3 + 3 (- x^{3}) + 3 \cdot - 3

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

- x^{6} + x^{3} \cdot - 3 + 3 x (- x^{3}) + 3 x \cdot - 3 + 3 (- x^{3}) + 3 \cdot - 3

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

Step 3.1.3

Move

- 3

$- 3$ to the left of

x^{3}

$x^{3}$ .

- x^{6} - 3 \cdot x^{3} + 3 x (- x^{3}) + 3 x \cdot - 3 + 3 (- x^{3}) + 3 \cdot - 3

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

- x^{6} - 3 x^{3} + 3 \cdot - 1 x \cdot x^{3} + 3 x \cdot - 3 + 3 (- x^{3}) + 3 \cdot - 3

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

Step 3.1.5

Multiply

x

$x$ by

x^{3}

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

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Step 3.1.5.1

Move

x^{3}

$x^{3}$ .

- x^{6} - 3 x^{3} + 3 \cdot - 1 (x^{3} x) + 3 x \cdot - 3 + 3 (- x^{3}) + 3 \cdot - 3

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

Step 3.1.5.2

Multiply

x^{3}

$x^{3}$ by

x

$x$ .

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Step 3.1.5.2.1

Raise

x

$x$ to the power of

1

$1$ .

- x^{6} - 3 x^{3} + 3 \cdot - 1 (x^{3} x^{1}) + 3 x \cdot - 3 + 3 (- x^{3}) + 3 \cdot - 3

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

Step 3.1.5.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

- x^{6} - 3 x^{3} + 3 \cdot - 1 x^{3 + 1} + 3 x \cdot - 3 + 3 (- x^{3}) + 3 \cdot - 3

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

- x^{6} - 3 x^{3} + 3 \cdot - 1 x^{3 + 1} + 3 x \cdot - 3 + 3 (- x^{3}) + 3 \cdot - 3

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

Step 3.1.5.3

Add

3

$3$ and

1

$1$ .

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

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

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

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

Step 3.1.6

Multiply

3

$3$ by

- 1

$- 1$ .

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

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

Step 3.1.7

Multiply

- 3

$- 3$ by

3

$3$ .

- x^{6} - 3 x^{3} - 3 x^{4} - 9 x + 3 (- x^{3}) + 3 \cdot - 3

$- x^{6} - 3 x^{3} - 3 x^{4} - 9 x + 3 (- x^{3}) + 3 \cdot - 3$

Step 3.1.8

Multiply

- 1

$- 1$ by

3

$3$ .

- x^{6} - 3 x^{3} - 3 x^{4} - 9 x - 3 x^{3} + 3 \cdot - 3

$- x^{6} - 3 x^{3} - 3 x^{4} - 9 x - 3 x^{3} + 3 \cdot - 3$

Step 3.1.9

Multiply

3

$3$ by

- 3

$- 3$ .

- x^{6} - 3 x^{3} - 3 x^{4} - 9 x - 3 x^{3} - 9

$- x^{6} - 3 x^{3} - 3 x^{4} - 9 x - 3 x^{3} - 9$

- x^{6} - 3 x^{3} - 3 x^{4} - 9 x - 3 x^{3} - 9

$- x^{6} - 3 x^{3} - 3 x^{4} - 9 x - 3 x^{3} - 9$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Subtract

3 x^{3}

$3 x^{3}$ from

- 3 x^{3}

$- 3 x^{3}$ .

- x^{6} - 6 x^{3} - 3 x^{4} - 9 x - 9

$- x^{6} - 6 x^{3} - 3 x^{4} - 9 x - 9$

Step 3.2.2

Move

- 6 x^{3}

$- 6 x^{3}$ .

- x^{6} - 3 x^{4} - 6 x^{3} - 9 x - 9

$- x^{6} - 3 x^{4} - 6 x^{3} - 9 x - 9$

- x^{6} - 3 x^{4} - 6 x^{3} - 9 x - 9

$- x^{6} - 3 x^{4} - 6 x^{3} - 9 x - 9$

- x^{6} - 3 x^{4} - 6 x^{3} - 9 x - 9

$- x^{6} - 3 x^{4} - 6 x^{3} - 9 x - 9$

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