Algebra Examples

$(3 x - 2) (3 x + 1) (3 x + 2) (x - 1)$

Step 1

Expand $(3 x - 2) (3 x + 1)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$(3 x (3 x + 1) - 2 (3 x + 1)) (3 x + 2) (x - 1)$

Step 1.2

Apply the distributive property.

$(3 x (3 x) + 3 x \cdot 1 - 2 (3 x + 1)) (3 x + 2) (x - 1)$

Step 1.3

Apply the distributive property.

$(3 x (3 x) + 3 x \cdot 1 - 2 (3 x) - 2 \cdot 1) (3 x + 2) (x - 1)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$(3 \cdot 3 x \cdot x + 3 x \cdot 1 - 2 (3 x) - 2 \cdot 1) (3 x + 2) (x - 1)$

Step 2.1.2

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

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Step 2.1.2.1

Move $x$ .

$(3 \cdot 3 (x \cdot x) + 3 x \cdot 1 - 2 (3 x) - 2 \cdot 1) (3 x + 2) (x - 1)$

Step 2.1.2.2

Multiply $x$ by $x$ .

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

Step 2.1.3

Multiply $3$ by $3$ .

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

Step 2.1.4

Multiply $3$ by $1$ .

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

Step 2.1.5

Multiply $3$ by $- 2$ .

$(9 x^{2} + 3 x - 6 x - 2 \cdot 1) (3 x + 2) (x - 1)$

Step 2.1.6

Multiply $- 2$ by $1$ .

$(9 x^{2} + 3 x - 6 x - 2) (3 x + 2) (x - 1)$

Step 2.2

Subtract $6 x$ from $3 x$ .

$(9 x^{2} - 3 x - 2) (3 x + 2) (x - 1)$

Step 3

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

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

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.2

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

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Step 4.1.2.1

Move $x$ .

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

Step 4.1.2.2

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

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Step 4.1.2.2.1

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

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

Step 4.1.2.2.2

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

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

Step 4.1.2.3

Add $1$ and $2$ .

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

Step 4.1.3

Multiply $9$ by $3$ .

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

Step 4.1.4

Multiply $2$ by $9$ .

$(27 x^{3} + 18 x^{2} - 3 x (3 x) - 3 x \cdot 2 - 2 (3 x) - 2 \cdot 2) (x - 1)$

Step 4.1.5

Rewrite using the commutative property of multiplication.

$(27 x^{3} + 18 x^{2} - 3 \cdot 3 x \cdot x - 3 x \cdot 2 - 2 (3 x) - 2 \cdot 2) (x - 1)$

Step 4.1.6

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

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Step 4.1.6.1

Move $x$ .

$(27 x^{3} + 18 x^{2} - 3 \cdot 3 (x \cdot x) - 3 x \cdot 2 - 2 (3 x) - 2 \cdot 2) (x - 1)$

Step 4.1.6.2

Multiply $x$ by $x$ .

$(27 x^{3} + 18 x^{2} - 3 \cdot 3 x^{2} - 3 x \cdot 2 - 2 (3 x) - 2 \cdot 2) (x - 1)$

Step 4.1.7

Multiply $- 3$ by $3$ .

$(27 x^{3} + 18 x^{2} - 9 x^{2} - 3 x \cdot 2 - 2 (3 x) - 2 \cdot 2) (x - 1)$

Step 4.1.8

Multiply $2$ by $- 3$ .

$(27 x^{3} + 18 x^{2} - 9 x^{2} - 6 x - 2 (3 x) - 2 \cdot 2) (x - 1)$

Step 4.1.9

Multiply $3$ by $- 2$ .

$(27 x^{3} + 18 x^{2} - 9 x^{2} - 6 x - 6 x - 2 \cdot 2) (x - 1)$

Step 4.1.10

Multiply $- 2$ by $2$ .

$(27 x^{3} + 18 x^{2} - 9 x^{2} - 6 x - 6 x - 4) (x - 1)$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Subtract $9 x^{2}$ from $18 x^{2}$ .

$(27 x^{3} + 9 x^{2} - 6 x - 6 x - 4) (x - 1)$

Step 4.2.2

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

$(27 x^{3} + 9 x^{2} - 12 x - 4) (x - 1)$

Step 5

Expand $(27 x^{3} + 9 x^{2} - 12 x - 4) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$27 x^{3} x + 27 x^{3} \cdot - 1 + 9 x^{2} x + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6

Simplify terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

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Step 6.1.1.1

Move $x$ .

$27 (x \cdot x^{3}) + 27 x^{3} \cdot - 1 + 9 x^{2} x + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.1.2

Multiply $x$ by $x^{3}$ .

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Step 6.1.1.2.1

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

$27 (x^{1} x^{3}) + 27 x^{3} \cdot - 1 + 9 x^{2} x + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.1.2.2

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

$27 x^{1 + 3} + 27 x^{3} \cdot - 1 + 9 x^{2} x + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.1.3

Add $1$ and $3$ .

$27 x^{4} + 27 x^{3} \cdot - 1 + 9 x^{2} x + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.2

Multiply $- 1$ by $27$ .

$27 x^{4} - 27 x^{3} + 9 x^{2} x + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.3

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

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Step 6.1.3.1

Move $x$ .

$27 x^{4} - 27 x^{3} + 9 (x \cdot x^{2}) + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.3.2

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

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Step 6.1.3.2.1

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

$27 x^{4} - 27 x^{3} + 9 (x^{1} x^{2}) + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.3.2.2

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

$27 x^{4} - 27 x^{3} + 9 x^{1 + 2} + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.3.3

Add $1$ and $2$ .

$27 x^{4} - 27 x^{3} + 9 x^{3} + 9 x^{2} \cdot - 1 - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.4

Multiply $- 1$ by $9$ .

$27 x^{4} - 27 x^{3} + 9 x^{3} - 9 x^{2} - 12 x \cdot x - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.5

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

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Step 6.1.5.1

Move $x$ .

$27 x^{4} - 27 x^{3} + 9 x^{3} - 9 x^{2} - 12 (x \cdot x) - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.5.2

Multiply $x$ by $x$ .

$27 x^{4} - 27 x^{3} + 9 x^{3} - 9 x^{2} - 12 x^{2} - 12 x \cdot - 1 - 4 x - 4 \cdot - 1$

Step 6.1.6

Multiply $- 1$ by $- 12$ .

$27 x^{4} - 27 x^{3} + 9 x^{3} - 9 x^{2} - 12 x^{2} + 12 x - 4 x - 4 \cdot - 1$

Step 6.1.7

Multiply $- 4$ by $- 1$ .

$27 x^{4} - 27 x^{3} + 9 x^{3} - 9 x^{2} - 12 x^{2} + 12 x - 4 x + 4$

Step 6.2

Simplify by adding terms.

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Step 6.2.1

Add $- 27 x^{3}$ and $9 x^{3}$ .

$27 x^{4} - 18 x^{3} - 9 x^{2} - 12 x^{2} + 12 x - 4 x + 4$

Step 6.2.2

Subtract $12 x^{2}$ from $- 9 x^{2}$ .

$27 x^{4} - 18 x^{3} - 21 x^{2} + 12 x - 4 x + 4$

Step 6.2.3

Subtract $4 x$ from $12 x$ .

$27 x^{4} - 18 x^{3} - 21 x^{2} + 8 x + 4$