Trigonometry Examples

$x^{3} - 27 = (x - 3) (x^{2} + 3 x + 9)$

Step 1

Expand $(x - 3) (x^{2} + 3 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} - 27 = x \cdot x^{2} + x (3 x) + x \cdot 9 - 3 x^{2} - 3 (3 x) - 3 \cdot 9$

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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

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Step 2.1.1.1

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

$x^{3} - 27 = x^{1} x^{2} + x (3 x) + x \cdot 9 - 3 x^{2} - 3 (3 x) - 3 \cdot 9$

Step 2.1.1.2

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

$x^{3} - 27 = x^{1 + 2} + x (3 x) + x \cdot 9 - 3 x^{2} - 3 (3 x) - 3 \cdot 9$

Step 2.1.2

Add $1$ and $2$ .

$x^{3} - 27 = x^{3} + x (3 x) + x \cdot 9 - 3 x^{2} - 3 (3 x) - 3 \cdot 9$

Step 2.2

Rewrite using the commutative property of multiplication.

$x^{3} - 27 = x^{3} + 3 x \cdot x + x \cdot 9 - 3 x^{2} - 3 (3 x) - 3 \cdot 9$

Step 2.3

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

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Step 2.3.1

Move $x$ .

$x^{3} - 27 = x^{3} + 3 (x \cdot x) + x \cdot 9 - 3 x^{2} - 3 (3 x) - 3 \cdot 9$

Step 2.3.2

Multiply $x$ by $x$ .

$x^{3} - 27 = x^{3} + 3 x^{2} + x \cdot 9 - 3 x^{2} - 3 (3 x) - 3 \cdot 9$

Step 2.4

Move $9$ to the left of $x$ .

$x^{3} - 27 = x^{3} + 3 x^{2} + 9 \cdot x - 3 x^{2} - 3 (3 x) - 3 \cdot 9$

Step 2.5

Multiply $3$ by $- 3$ .

$x^{3} - 27 = x^{3} + 3 x^{2} + 9 x - 3 x^{2} - 9 x - 3 \cdot 9$

Step 2.6

Multiply $- 3$ by $9$ .

$x^{3} - 27 = x^{3} + 3 x^{2} + 9 x - 3 x^{2} - 9 x - 27$

Step 3

Combine the opposite terms in $x^{3} + 3 x^{2} + 9 x - 3 x^{2} - 9 x - 27$ .

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Step 3.1

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

$x^{3} - 27 = x^{3} + 9 x + 0 - 9 x - 27$

Step 3.2

Add $x^{3} + 9 x$ and $0$ .

$x^{3} - 27 = x^{3} + 9 x - 9 x - 27$

Step 3.3

Subtract $9 x$ from $9 x$ .

$x^{3} - 27 = x^{3} + 0 - 27$

Step 3.4

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

$x^{3} - 27 = x^{3} - 27$

Step 4

Since the two sides have been shown to be equivalent, the equation is an identity.

$x^{3} - 27 = (x - 3) (x^{2} + 3 x + 9)$ is an identity.