Algebra Examples

$\sqrt[4]{256 {(x^{2} - 1)}^{12}}$

Step 1

Rewrite $1$ as $1^{2}$ .

$\sqrt[4]{256 {(x^{2} - 1^{2})}^{12}}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\sqrt[4]{256 {((x + 1) (x - 1))}^{12}}$

Step 3

Apply the product rule to $(x + 1) (x - 1)$ .

$\sqrt[4]{256 {(x + 1)}^{12} {(x - 1)}^{12}}$

Step 4

Rewrite $256 {(x + 1)}^{12} {(x - 1)}^{12}$ as ${(4 {(x + 1)}^{3} {(x - 1)}^{3})}^{4}$ .

$\sqrt[4]{{(4 {(x + 1)}^{3} {(x - 1)}^{3})}^{4}}$

Step 5

Pull terms out from under the radical.

$| 4 {(x + 1)}^{3} {(x - 1)}^{3} |$

Step 6

Use the Binomial Theorem.

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

Step 7

Simplify each term.

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Step 7.1

Multiply $3$ by $1$ .

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

Step 7.2

One to any power is one.

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

Step 7.3

Multiply $3$ by $1$ .

$| 4 (x^{3} + 3 x^{2} + 3 x + 1^{3}) {(x - 1)}^{3} |$

Step 7.4

One to any power is one.

$| 4 (x^{3} + 3 x^{2} + 3 x + 1) {(x - 1)}^{3} |$

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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Step 9.1

Multiply $3$ by $4$ .

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

Step 9.2

Multiply $3$ by $4$ .

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

Step 9.3

Multiply $4$ by $1$ .

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

Step 10

Use the Binomial Theorem.

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

Step 11

Simplify each term.

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Step 11.1

Multiply $- 1$ by $3$ .

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

Step 11.2

Raise $- 1$ to the power of $2$ .

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

Step 11.3

Multiply $3$ by $1$ .

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

Step 11.4

Raise $- 1$ to the power of $3$ .

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

Step 12

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

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

Step 13

Combine the opposite terms in $4 x^{3} x^{3} + 4 x^{3} (- 3 x^{2}) + 4 x^{3} (3 x) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} (3 x) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (- 3 x^{2}) + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1$ .

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Step 13.1

Reorder the factors in the terms $12 x^{2} (3 x)$ and $12 x (- 3 x^{2})$ .

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

Step 13.2

Subtract $3 \cdot 12 x^{2} x$ from $3 \cdot 12 x^{2} x$ .

$| 4 x^{3} x^{3} + 4 x^{3} (- 3 x^{2}) + 4 x^{3} (3 x) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 0 + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 13.3

Add $4 x^{3} x^{3} + 4 x^{3} (- 3 x^{2}) + 4 x^{3} (3 x) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3}$ and $0$ .

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

Step 14

Simplify each term.

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Step 14.1

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

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Step 14.1.1

Move $x^{3}$ .

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

Step 14.1.2

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

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

Step 14.1.3

Add $3$ and $3$ .

$| 4 x^{6} + 4 x^{3} (- 3 x^{2}) + 4 x^{3} (3 x) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.2

Rewrite using the commutative property of multiplication.

$| 4 x^{6} + 4 \cdot - 3 x^{3} x^{2} + 4 x^{3} (3 x) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.3

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 14.3.1

Move $x^{2}$ .

$| 4 x^{6} + 4 \cdot - 3 (x^{2} x^{3}) + 4 x^{3} (3 x) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.3.2

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

$| 4 x^{6} + 4 \cdot - 3 x^{2 + 3} + 4 x^{3} (3 x) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.3.3

Add $2$ and $3$ .

$| 4 x^{6} + 4 \cdot - 3 x^{5} + 4 x^{3} (3 x) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.4

Multiply $4$ by $- 3$ .

$| 4 x^{6} - 12 x^{5} + 4 x^{3} (3 x) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.5

Rewrite using the commutative property of multiplication.

$| 4 x^{6} - 12 x^{5} + 4 \cdot 3 x^{3} x + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.6

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

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Step 14.6.1

Move $x$ .

$| 4 x^{6} - 12 x^{5} + 4 \cdot 3 (x \cdot x^{3}) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.6.2

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

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Step 14.6.2.1

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

$| 4 x^{6} - 12 x^{5} + 4 \cdot 3 (x^{1} x^{3}) + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.6.2.2

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

$| 4 x^{6} - 12 x^{5} + 4 \cdot 3 x^{1 + 3} + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.6.3

Add $1$ and $3$ .

$| 4 x^{6} - 12 x^{5} + 4 \cdot 3 x^{4} + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.7

Multiply $4$ by $3$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} + 4 x^{3} \cdot - 1 + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.8

Multiply $- 1$ by $4$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{2} x^{3} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.9

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 14.9.1

Move $x^{3}$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 (x^{3} x^{2}) + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.9.2

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

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{3 + 2} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.9.3

Add $3$ and $2$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} + 12 x^{2} (- 3 x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.10

Rewrite using the commutative property of multiplication.

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} + 12 \cdot - 3 x^{2} x^{2} + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.11

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

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Step 14.11.1

Move $x^{2}$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} + 12 \cdot - 3 (x^{2} x^{2}) + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.11.2

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

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} + 12 \cdot - 3 x^{2 + 2} + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.11.3

Add $2$ and $2$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} + 12 \cdot - 3 x^{4} + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.12

Multiply $12$ by $- 3$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} + 12 x^{2} \cdot - 1 + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.13

Multiply $- 1$ by $12$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x \cdot x^{3} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.14

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

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Step 14.14.1

Move $x^{3}$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 (x^{3} x) + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.14.2

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

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Step 14.14.2.1

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

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 (x^{3} x^{1}) + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.14.2.2

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

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{3 + 1} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.14.3

Add $3$ and $1$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 12 x (3 x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.15

Rewrite using the commutative property of multiplication.

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 12 \cdot 3 x \cdot x + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.16

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

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Step 14.16.1

Move $x$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 12 \cdot 3 (x \cdot x) + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.16.2

Multiply $x$ by $x$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 12 \cdot 3 x^{2} + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.17

Multiply $12$ by $3$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} + 12 x \cdot - 1 + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.18

Multiply $- 1$ by $12$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x + 4 x^{3} + 4 (- 3 x^{2}) + 4 (3 x) + 4 \cdot - 1 |$

Step 14.19

Multiply $- 3$ by $4$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x + 4 x^{3} - 12 x^{2} + 4 (3 x) + 4 \cdot - 1 |$

Step 14.20

Multiply $3$ by $4$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x + 4 x^{3} - 12 x^{2} + 12 x + 4 \cdot - 1 |$

Step 14.21

Multiply $4$ by $- 1$ .

$| 4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x + 4 x^{3} - 12 x^{2} + 12 x - 4 |$

Step 15

Combine the opposite terms in $4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3} + 12 x^{5} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x + 4 x^{3} - 12 x^{2} + 12 x - 4$ .

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Step 15.1

Add $- 12 x^{5}$ and $12 x^{5}$ .

$| 4 x^{6} + 12 x^{4} - 4 x^{3} + 0 - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x + 4 x^{3} - 12 x^{2} + 12 x - 4 |$

Step 15.2

Add $4 x^{6} + 12 x^{4} - 4 x^{3}$ and $0$ .

$| 4 x^{6} + 12 x^{4} - 4 x^{3} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x + 4 x^{3} - 12 x^{2} + 12 x - 4 |$

Step 15.3

Add $- 4 x^{3}$ and $4 x^{3}$ .

$| 4 x^{6} + 12 x^{4} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x + 0 - 12 x^{2} + 12 x - 4 |$

Step 15.4

Add $4 x^{6} + 12 x^{4} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x$ and $0$ .

$| 4 x^{6} + 12 x^{4} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x - 12 x^{2} + 12 x - 4 |$

Step 15.5

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

$| 4 x^{6} + 12 x^{4} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x^{2} + 0 - 4 |$

Step 15.6

Add $4 x^{6} + 12 x^{4} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x^{2}$ and $0$ .

$| 4 x^{6} + 12 x^{4} - 36 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x^{2} - 4 |$

Step 16

Subtract $36 x^{4}$ from $12 x^{4}$ .

$| 4 x^{6} - 24 x^{4} - 12 x^{2} + 12 x^{4} + 36 x^{2} - 12 x^{2} - 4 |$

Step 17

Add $- 24 x^{4}$ and $12 x^{4}$ .

$| 4 x^{6} - 12 x^{4} - 12 x^{2} + 36 x^{2} - 12 x^{2} - 4 |$

Step 18

Add $- 12 x^{2}$ and $36 x^{2}$ .

$| 4 x^{6} - 12 x^{4} + 24 x^{2} - 12 x^{2} - 4 |$

Step 19

Subtract $12 x^{2}$ from $24 x^{2}$ .

$| 4 x^{6} - 12 x^{4} + 12 x^{2} - 4 |$