Algebra Examples

${(3 x - 2)}^{3} - 27$

Step 1

Rewrite $27$ as $3^{3}$ .

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

Step 2

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

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Subtract $3$ from $- 2$ .

$(3 x - 5) ({(3 x - 2)}^{2} + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.2

Rewrite ${(3 x - 2)}^{2}$ as $(3 x - 2) (3 x - 2)$ .

$(3 x - 5) ((3 x - 2) (3 x - 2) + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.3

Expand $(3 x - 2) (3 x - 2)$ using the FOIL Method.

Tap for more steps...

Step 3.3.1

Apply the distributive property.

$(3 x - 5) (3 x (3 x - 2) - 2 (3 x - 2) + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.3.2

Apply the distributive property.

$(3 x - 5) (3 x (3 x) + 3 x \cdot - 2 - 2 (3 x - 2) + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.3.3

Apply the distributive property.

$(3 x - 5) (3 x (3 x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.4

Simplify and combine like terms.

Tap for more steps...

Step 3.4.1

Simplify each term.

Tap for more steps...

Step 3.4.1.1

Rewrite using the commutative property of multiplication.

$(3 x - 5) (3 \cdot 3 x \cdot x + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.4.1.2

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

Tap for more steps...

Step 3.4.1.2.1

Move $x$ .

$(3 x - 5) (3 \cdot 3 (x \cdot x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.4.1.2.2

Multiply $x$ by $x$ .

$(3 x - 5) (3 \cdot 3 x^{2} + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.4.1.3

Multiply $3$ by $3$ .

$(3 x - 5) (9 x^{2} + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2 + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.4.1.4

Multiply $- 2$ by $3$ .

$(3 x - 5) (9 x^{2} - 6 x - 2 (3 x) - 2 \cdot - 2 + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.4.1.5

Multiply $3$ by $- 2$ .

$(3 x - 5) (9 x^{2} - 6 x - 6 x - 2 \cdot - 2 + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.4.1.6

Multiply $- 2$ by $- 2$ .

$(3 x - 5) (9 x^{2} - 6 x - 6 x + 4 + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.4.2

Subtract $6 x$ from $- 6 x$ .

$(3 x - 5) (9 x^{2} - 12 x + 4 + (3 x - 2) \cdot 3 + 3^{2})$

Step 3.5

Apply the distributive property.

$(3 x - 5) (9 x^{2} - 12 x + 4 + 3 x \cdot 3 - 2 \cdot 3 + 3^{2})$

Step 3.6

Multiply $3$ by $3$ .

$(3 x - 5) (9 x^{2} - 12 x + 4 + 9 x - 2 \cdot 3 + 3^{2})$

Step 3.7

Multiply $- 2$ by $3$ .

$(3 x - 5) (9 x^{2} - 12 x + 4 + 9 x - 6 + 3^{2})$

Step 3.8

Raise $3$ to the power of $2$ .

$(3 x - 5) (9 x^{2} - 12 x + 4 + 9 x - 6 + 9)$

Step 3.9

Add $- 12 x$ and $9 x$ .

$(3 x - 5) (9 x^{2} - 3 x + 4 - 6 + 9)$

Step 3.10

Subtract $6$ from $4$ .

$(3 x - 5) (9 x^{2} - 3 x - 2 + 9)$

Step 3.11

Add $- 2$ and $9$ .

$(3 x - 5) (9 x^{2} - 3 x + 7)$