Algebra Examples

$f (x) = {(x - 3)}^{4} {(x + 6)}^{2}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $- 3$ by $4$ .

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

Step 2.1.2

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

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

Step 2.1.3

Multiply $9$ by $6$ .

$(x^{4} - 12 x^{3} + 54 x^{2} + 4 x {(- 3)}^{3} + {(- 3)}^{4}) {(x + 6)}^{2}$

Step 2.1.4

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

$(x^{4} - 12 x^{3} + 54 x^{2} + 4 x \cdot - 27 + {(- 3)}^{4}) {(x + 6)}^{2}$

Step 2.1.5

Multiply $- 27$ by $4$ .

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + {(- 3)}^{4}) {(x + 6)}^{2}$

Step 2.1.6

Raise $- 3$ to the power of $4$ .

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) {(x + 6)}^{2}$

Step 2.2

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) ((x + 6) (x + 6))$

Step 3

Expand $(x + 6) (x + 6)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) (x (x + 6) + 6 (x + 6))$

Step 3.2

Apply the distributive property.

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) (x \cdot x + x \cdot 6 + 6 (x + 6))$

Step 3.3

Apply the distributive property.

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) (x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6)$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $x$ by $x$ .

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) (x^{2} + x \cdot 6 + 6 x + 6 \cdot 6)$

Step 4.1.2

Move $6$ to the left of $x$ .

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) (x^{2} + 6 \cdot x + 6 x + 6 \cdot 6)$

Step 4.1.3

Multiply $6$ by $6$ .

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) (x^{2} + 6 x + 6 x + 36)$

Step 4.2

Add $6 x$ and $6 x$ .

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) (x^{2} + 12 x + 36)$

Step 5

Expand $(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) (x^{2} + 12 x + 36)$ by multiplying each term in the first expression by each term in the second expression.

$x^{4} x^{2} + x^{4} (12 x) + x^{4} \cdot 36 - 12 x^{3} x^{2} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6

Simplify terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 6.1.1.1

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

$x^{4 + 2} + x^{4} (12 x) + x^{4} \cdot 36 - 12 x^{3} x^{2} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.1.2

Add $4$ and $2$ .

$x^{6} + x^{4} (12 x) + x^{4} \cdot 36 - 12 x^{3} x^{2} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.2

Rewrite using the commutative property of multiplication.

$x^{6} + 12 x^{4} x + x^{4} \cdot 36 - 12 x^{3} x^{2} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.3

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 6.1.3.1

Move $x$ .

$x^{6} + 12 (x \cdot x^{4}) + x^{4} \cdot 36 - 12 x^{3} x^{2} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.3.2

Multiply $x$ by $x^{4}$ .

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Step 6.1.3.2.1

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

$x^{6} + 12 (x^{1} x^{4}) + x^{4} \cdot 36 - 12 x^{3} x^{2} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.3.2.2

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

$x^{6} + 12 x^{1 + 4} + x^{4} \cdot 36 - 12 x^{3} x^{2} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.3.3

Add $1$ and $4$ .

$x^{6} + 12 x^{5} + x^{4} \cdot 36 - 12 x^{3} x^{2} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.4

Move $36$ to the left of $x^{4}$ .

$x^{6} + 12 x^{5} + 36 \cdot x^{4} - 12 x^{3} x^{2} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.5

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

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Step 6.1.5.1

Move $x^{2}$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 (x^{2} x^{3}) - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.5.2

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

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{2 + 3} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.5.3

Add $2$ and $3$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 12 x^{3} (12 x) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.6

Rewrite using the commutative property of multiplication.

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 12 \cdot 12 x^{3} x - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.7

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

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Step 6.1.7.1

Move $x$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 12 \cdot 12 (x \cdot x^{3}) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.7.2

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

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Step 6.1.7.2.1

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

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 12 \cdot 12 (x^{1} x^{3}) - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.7.2.2

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

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 12 \cdot 12 x^{1 + 3} - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.7.3

Add $1$ and $3$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 12 \cdot 12 x^{4} - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.8

Multiply $- 12$ by $12$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 12 x^{3} \cdot 36 + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.9

Multiply $36$ by $- 12$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{2} x^{2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.10

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

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Step 6.1.10.1

Move $x^{2}$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 (x^{2} x^{2}) + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.10.2

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

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{2 + 2} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.10.3

Add $2$ and $2$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 54 x^{2} (12 x) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.11

Rewrite using the commutative property of multiplication.

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 54 \cdot 12 x^{2} x + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.12

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

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Step 6.1.12.1

Move $x$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 54 \cdot 12 (x \cdot x^{2}) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.12.2

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

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Step 6.1.12.2.1

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

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 54 \cdot 12 (x^{1} x^{2}) + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.12.2.2

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

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 54 \cdot 12 x^{1 + 2} + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.12.3

Add $1$ and $2$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 54 \cdot 12 x^{3} + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.13

Multiply $54$ by $12$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 54 x^{2} \cdot 36 - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.14

Multiply $36$ by $54$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x \cdot x^{2} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.15

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

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Step 6.1.15.1

Move $x^{2}$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 (x^{2} x) - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.15.2

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

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Step 6.1.15.2.1

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

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 (x^{2} x^{1}) - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.15.2.2

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

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{2 + 1} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.15.3

Add $2$ and $1$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 108 x (12 x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.16

Rewrite using the commutative property of multiplication.

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 108 \cdot 12 x \cdot x - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.17

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

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Step 6.1.17.1

Move $x$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 108 \cdot 12 (x \cdot x) - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.17.2

Multiply $x$ by $x$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 108 \cdot 12 x^{2} - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.18

Multiply $- 108$ by $12$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 108 x \cdot 36 + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.19

Multiply $36$ by $- 108$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 3888 x + 81 x^{2} + 81 (12 x) + 81 \cdot 36$

Step 6.1.20

Multiply $12$ by $81$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 3888 x + 81 x^{2} + 972 x + 81 \cdot 36$

Step 6.1.21

Multiply $81$ by $36$ .

$x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 3888 x + 81 x^{2} + 972 x + 2916$

Step 6.2

Simplify by adding terms.

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Step 6.2.1

Combine the opposite terms in $x^{6} + 12 x^{5} + 36 x^{4} - 12 x^{5} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 3888 x + 81 x^{2} + 972 x + 2916$ .

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Step 6.2.1.1

Subtract $12 x^{5}$ from $12 x^{5}$ .

$x^{6} + 36 x^{4} + 0 - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 3888 x + 81 x^{2} + 972 x + 2916$

Step 6.2.1.2

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

$x^{6} + 36 x^{4} - 144 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 3888 x + 81 x^{2} + 972 x + 2916$

Step 6.2.2

Subtract $144 x^{4}$ from $36 x^{4}$ .

$x^{6} - 108 x^{4} - 432 x^{3} + 54 x^{4} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 3888 x + 81 x^{2} + 972 x + 2916$

Step 6.2.3

Add $- 108 x^{4}$ and $54 x^{4}$ .

$x^{6} - 54 x^{4} - 432 x^{3} + 648 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 3888 x + 81 x^{2} + 972 x + 2916$

Step 6.2.4

Add $- 432 x^{3}$ and $648 x^{3}$ .

$x^{6} - 54 x^{4} + 216 x^{3} + 1944 x^{2} - 108 x^{3} - 1296 x^{2} - 3888 x + 81 x^{2} + 972 x + 2916$

Step 6.2.5

Subtract $108 x^{3}$ from $216 x^{3}$ .

$x^{6} - 54 x^{4} + 108 x^{3} + 1944 x^{2} - 1296 x^{2} - 3888 x + 81 x^{2} + 972 x + 2916$

Step 6.2.6

Subtract $1296 x^{2}$ from $1944 x^{2}$ .

$x^{6} - 54 x^{4} + 108 x^{3} + 648 x^{2} - 3888 x + 81 x^{2} + 972 x + 2916$

Step 6.2.7

Add $648 x^{2}$ and $81 x^{2}$ .

$x^{6} - 54 x^{4} + 108 x^{3} + 729 x^{2} - 3888 x + 972 x + 2916$

Step 6.2.8

Add $- 3888 x$ and $972 x$ .

$x^{6} - 54 x^{4} + 108 x^{3} + 729 x^{2} - 2916 x + 2916$

Step 7

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 27, \pm 36, \pm 54, \pm 81, \pm 108, \pm 162, \pm 243, \pm 324, \pm 486, \pm 729, \pm 972, \pm 1458, \pm 2916$

$q = \pm 1$

Step 8

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 27, \pm 36, \pm 54, \pm 81, \pm 108, \pm 162, \pm 243, \pm 324, \pm 486, \pm 729, \pm 972, \pm 1458, \pm 2916$

Step 9

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(3)}^{6} - 54 {(3)}^{4} + 108 {(3)}^{3} + 729 {(3)}^{2} - 2916 \cdot 3 + 2916$

Step 10

Simplify the expression. In this case, the expression is equal to $0$ so $x = 3$ is a root of the polynomial.

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Step 10.1

Simplify each term.

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Step 10.1.1

Raise $3$ to the power of $6$ .

$729 - 54 {(3)}^{4} + 108 {(3)}^{3} + 729 {(3)}^{2} - 2916 \cdot 3 + 2916$

Step 10.1.2

Raise $3$ to the power of $4$ .

$729 - 54 \cdot 81 + 108 {(3)}^{3} + 729 {(3)}^{2} - 2916 \cdot 3 + 2916$

Step 10.1.3

Multiply $- 54$ by $81$ .

$729 - 4374 + 108 {(3)}^{3} + 729 {(3)}^{2} - 2916 \cdot 3 + 2916$

Step 10.1.4

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

$729 - 4374 + 108 \cdot 27 + 729 {(3)}^{2} - 2916 \cdot 3 + 2916$

Step 10.1.5

Multiply $108$ by $27$ .

$729 - 4374 + 2916 + 729 {(3)}^{2} - 2916 \cdot 3 + 2916$

Step 10.1.6

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

$729 - 4374 + 2916 + 729 \cdot 9 - 2916 \cdot 3 + 2916$

Step 10.1.7

Multiply $729$ by $9$ .

$729 - 4374 + 2916 + 6561 - 2916 \cdot 3 + 2916$

Step 10.1.8

Multiply $- 2916$ by $3$ .

$729 - 4374 + 2916 + 6561 - 8748 + 2916$

Step 10.2

Simplify by adding and subtracting.

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Step 10.2.1

Subtract $4374$ from $729$ .

$- 3645 + 2916 + 6561 - 8748 + 2916$

Step 10.2.2

Add $- 3645$ and $2916$ .

$- 729 + 6561 - 8748 + 2916$

Step 10.2.3

Add $- 729$ and $6561$ .

$5832 - 8748 + 2916$

Step 10.2.4

Subtract $8748$ from $5832$ .

$- 2916 + 2916$

Step 10.2.5

Add $- 2916$ and $2916$ .

$0$

Step 11

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{6} - 54 x^{4} + 108 x^{3} + 729 x^{2} - 2916 x + 2916}{x - 3}$

Step 12

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 12.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$

Step 12.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$

	$1$

Step 12.3

Multiply the newest entry in the result $(1)$ by the divisor $(3)$ and place the result of $(3)$ under the next term in the dividend $(0)$ .

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$
	$1$

Step 12.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$
	$1$	$3$

Step 12.5

Multiply the newest entry in the result $(3)$ by the divisor $(3)$ and place the result of $(9)$ under the next term in the dividend $(- 54)$ .

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$
	$1$	$3$

Step 12.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$
	$1$	$3$	$- 45$

Step 12.7

Multiply the newest entry in the result $(- 45)$ by the divisor $(3)$ and place the result of $(- 135)$ under the next term in the dividend $(108)$ .

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$	$- 135$
	$1$	$3$	$- 45$

Step 12.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$	$- 135$
	$1$	$3$	$- 45$	$- 27$

Step 12.9

Multiply the newest entry in the result $(- 27)$ by the divisor $(3)$ and place the result of $(- 81)$ under the next term in the dividend $(729)$ .

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$	$- 135$	$- 81$
	$1$	$3$	$- 45$	$- 27$

Step 12.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$	$- 135$	$- 81$
	$1$	$3$	$- 45$	$- 27$	$648$

Step 12.11

Multiply the newest entry in the result $(648)$ by the divisor $(3)$ and place the result of $(1944)$ under the next term in the dividend $(- 2916)$ .

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$	$- 135$	$- 81$	$1944$
	$1$	$3$	$- 45$	$- 27$	$648$

Step 12.12

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$	$- 135$	$- 81$	$1944$
	$1$	$3$	$- 45$	$- 27$	$648$	$- 972$

Step 12.13

Multiply the newest entry in the result $(- 972)$ by the divisor $(3)$ and place the result of $(- 2916)$ under the next term in the dividend $(2916)$ .

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$	$- 135$	$- 81$	$1944$	$- 2916$
	$1$	$3$	$- 45$	$- 27$	$648$	$- 972$

Step 12.14

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 54$	$108$	$729$	$- 2916$	$2916$
		$3$	$9$	$- 135$	$- 81$	$1944$	$- 2916$
	$1$	$3$	$- 45$	$- 27$	$648$	$- 972$	$0$

Step 12.15

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{5} + 3 x^{4} - 45 x^{3} + - 27 x^{2} + (648) x - 972$

Step 12.16

Simplify the quotient polynomial.

$x^{5} + 3 x^{4} - 45 x^{3} - 27 x^{2} + 648 x - 972$

Step 13

Solve the equation to find any remaining roots.

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Step 13.1

Factor the left side of the equation.

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Step 13.1.1

Regroup terms.

$x^{5} - 27 x^{2} + 3 x^{4} - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.2

Factor $x^{2}$ out of $x^{5} - 27 x^{2}$ .

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Step 13.1.2.1

Factor $x^{2}$ out of $x^{5}$ .

$x^{2} x^{3} - 27 x^{2} + 3 x^{4} - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.2.2

Factor $x^{2}$ out of $- 27 x^{2}$ .

$x^{2} x^{3} + x^{2} \cdot - 27 + 3 x^{4} - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.2.3

Factor $x^{2}$ out of $x^{2} x^{3} + x^{2} \cdot - 27$ .

$x^{2} (x^{3} - 27) + 3 x^{4} - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.3

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

$x^{2} (x^{3} - 3^{3}) + 3 x^{4} - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.4

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 $a = x$ and $b = 3$ .

$x^{2} ((x - 3) (x^{2} + x \cdot 3 + 3^{2})) + 3 x^{4} - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.5

Factor.

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Step 13.1.5.1

Simplify.

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Step 13.1.5.1.1

Move $3$ to the left of $x$ .

$x^{2} ((x - 3) (x^{2} + 3 x + 3^{2})) + 3 x^{4} - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.5.1.2

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

$x^{2} ((x - 3) (x^{2} + 3 x + 9)) + 3 x^{4} - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.5.2

Remove unnecessary parentheses.

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 x^{4} - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.6

Factor $3$ out of $3 x^{4} - 45 x^{3} + 648 x - 972$ .

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Step 13.1.6.1

Factor $3$ out of $3 x^{4}$ .

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 (x^{4}) - 45 x^{3} + 648 x - 972 = 0$

Step 13.1.6.2

Factor $3$ out of $- 45 x^{3}$ .

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 (x^{4}) + 3 (- 15 x^{3}) + 648 x - 972 = 0$

Step 13.1.6.3

Factor $3$ out of $648 x$ .

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 (x^{4}) + 3 (- 15 x^{3}) + 3 (216 x) - 972 = 0$

Step 13.1.6.4

Factor $3$ out of $- 972$ .

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 x^{4} + 3 (- 15 x^{3}) + 3 (216 x) + 3 \cdot - 324 = 0$

Step 13.1.6.5

Factor $3$ out of $3 x^{4} + 3 (- 15 x^{3})$ .

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 (x^{4} - 15 x^{3}) + 3 (216 x) + 3 \cdot - 324 = 0$

Step 13.1.6.6

Factor $3$ out of $3 (x^{4} - 15 x^{3}) + 3 (216 x)$ .

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 (x^{4} - 15 x^{3} + 216 x) + 3 \cdot - 324 = 0$

Step 13.1.6.7

Factor $3$ out of $3 (x^{4} - 15 x^{3} + 216 x) + 3 \cdot - 324$ .

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 (x^{4} - 15 x^{3} + 216 x - 324) = 0$

Step 13.1.7

Factor.

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Step 13.1.7.1

Factor $x^{4} - 15 x^{3} + 216 x - 324$ using the rational roots test.

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Step 13.1.7.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 324, \pm 2, \pm 162, \pm 3, \pm 108, \pm 4, \pm 81, \pm 6, \pm 54, \pm 9, \pm 36, \pm 12, \pm 27, \pm 18$

$q = \pm 1$

Step 13.1.7.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 324, \pm 2, \pm 162, \pm 3, \pm 108, \pm 4, \pm 81, \pm 6, \pm 54, \pm 9, \pm 36, \pm 12, \pm 27, \pm 18$

Step 13.1.7.1.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 13.1.7.1.3.1

Substitute $3$ into the polynomial.

$3^{4} - 15 \cdot 3^{3} + 216 \cdot 3 - 324$

Step 13.1.7.1.3.2

Raise $3$ to the power of $4$ .

$81 - 15 \cdot 3^{3} + 216 \cdot 3 - 324$

Step 13.1.7.1.3.3

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

$81 - 15 \cdot 27 + 216 \cdot 3 - 324$

Step 13.1.7.1.3.4

Multiply $- 15$ by $27$ .

$81 - 405 + 216 \cdot 3 - 324$

Step 13.1.7.1.3.5

Subtract $405$ from $81$ .

$- 324 + 216 \cdot 3 - 324$

Step 13.1.7.1.3.6

Multiply $216$ by $3$ .

$- 324 + 648 - 324$

Step 13.1.7.1.3.7

Add $- 324$ and $648$ .

$324 - 324$

Step 13.1.7.1.3.8

Subtract $324$ from $324$ .

$0$

Step 13.1.7.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 15 x^{3} + 216 x - 324}{x - 3}$

Step 13.1.7.1.5

Divide $x^{4} - 15 x^{3} + 216 x - 324$ by $x - 3$ .

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Step 13.1.7.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

Step 13.1.7.1.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

Step 13.1.7.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

Step 13.1.7.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 3 x^{3}$

$x^{3}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

Step 13.1.7.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

Step 13.1.7.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

Step 13.1.7.1.5.7

Divide the highest order term in the dividend $- 12 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

Step 13.1.7.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

Step 13.1.7.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{3} + 36 x^{2}$

$x^{3}$

$12 x^{2}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

Step 13.1.7.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

Step 13.1.7.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$12 x^{2}$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

Step 13.1.7.1.5.12

Divide the highest order term in the dividend $- 36 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$36 x$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

Step 13.1.7.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$36 x$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

$36 x^{2}$

$108 x$

Step 13.1.7.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 36 x^{2} + 108 x$

$x^{3}$

$12 x^{2}$

$36 x$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

$36 x^{2}$

$108 x$

Step 13.1.7.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$36 x$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

$36 x^{2}$

$108 x$

Step 13.1.7.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$12 x^{2}$

$36 x$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

$36 x^{2}$

$108 x$

$324$

Step 13.1.7.1.5.17

Divide the highest order term in the dividend $108 x$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$36 x$

$108$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

$36 x^{2}$

$108 x$

$324$

Step 13.1.7.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$36 x$

$108$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

$36 x^{2}$

$108 x$

$324$

$108 x$

$324$

Step 13.1.7.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $108 x - 324$

$x^{3}$

$12 x^{2}$

$36 x$

$108$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

$36 x^{2}$

$108 x$

$324$

$108 x$

$324$

Step 13.1.7.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$36 x$

$108$

$x$

$3$

$x^{4}$

$15 x^{3}$

$0 x^{2}$

$216 x$

$324$

$x^{4}$

$3 x^{3}$

$12 x^{3}$

$0 x^{2}$

$12 x^{3}$

$36 x^{2}$

$216 x$

$36 x^{2}$

$108 x$

$324$

$108 x$

$324$

$0$

Step 13.1.7.1.5.21

Since the remander is $0$ , the final answer is the quotient.

$x^{3} - 12 x^{2} - 36 x + 108$

Step 13.1.7.1.6

Write $x^{4} - 15 x^{3} + 216 x - 324$ as a set of factors.

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 ((x - 3) (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.7.2

Remove unnecessary parentheses.

$x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 (x - 3) (x^{3} - 12 x^{2} - 36 x + 108) = 0$

Step 13.1.8

Factor $x - 3$ out of $x^{2} (x - 3) (x^{2} + 3 x + 9) + 3 (x - 3) (x^{3} - 12 x^{2} - 36 x + 108)$ .

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Step 13.1.8.1

Factor $x - 3$ out of $x^{2} (x - 3) (x^{2} + 3 x + 9)$ .

$(x - 3) (x^{2} (x^{2} + 3 x + 9)) + 3 (x - 3) (x^{3} - 12 x^{2} - 36 x + 108) = 0$

Step 13.1.8.2

Factor $x - 3$ out of $3 (x - 3) (x^{3} - 12 x^{2} - 36 x + 108)$ .

$(x - 3) (x^{2} (x^{2} + 3 x + 9)) + (x - 3) (3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.8.3

Factor $x - 3$ out of $(x - 3) (x^{2} (x^{2} + 3 x + 9)) + (x - 3) (3 (x^{3} - 12 x^{2} - 36 x + 108))$ .

$(x - 3) (x^{2} (x^{2} + 3 x + 9) + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.9

Apply the distributive property.

$(x - 3) (x^{2} x^{2} + x^{2} (3 x) + x^{2} \cdot 9 + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.10

Simplify.

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Step 13.1.10.1

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

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Step 13.1.10.1.1

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

$(x - 3) (x^{2 + 2} + x^{2} (3 x) + x^{2} \cdot 9 + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.10.1.2

Add $2$ and $2$ .

$(x - 3) (x^{4} + x^{2} (3 x) + x^{2} \cdot 9 + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.10.2

Rewrite using the commutative property of multiplication.

$(x - 3) (x^{4} + 3 x^{2} x + x^{2} \cdot 9 + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.10.3

Move $9$ to the left of $x^{2}$ .

$(x - 3) (x^{4} + 3 x^{2} x + 9 \cdot x^{2} + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.11

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

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Step 13.1.11.1

Move $x$ .

$(x - 3) (x^{4} + 3 (x \cdot x^{2}) + 9 \cdot x^{2} + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.11.2

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

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Step 13.1.11.2.1

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

$(x - 3) (x^{4} + 3 (x \cdot x^{2}) + 9 \cdot x^{2} + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.11.2.2

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

$(x - 3) (x^{4} + 3 x^{1 + 2} + 9 \cdot x^{2} + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.11.3

Add $1$ and $2$ .

$(x - 3) (x^{4} + 3 x^{3} + 9 \cdot x^{2} + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

$(x - 3) (x^{4} + 3 x^{3} + 9 x^{2} + 3 (x^{3} - 12 x^{2} - 36 x + 108)) = 0$

Step 13.1.12

Apply the distributive property.

$(x - 3) (x^{4} + 3 x^{3} + 9 x^{2} + 3 x^{3} + 3 (- 12 x^{2}) + 3 (- 36 x) + 3 \cdot 108) = 0$

Step 13.1.13

Simplify.

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Step 13.1.13.1

Multiply $- 12$ by $3$ .

$(x - 3) (x^{4} + 3 x^{3} + 9 x^{2} + 3 x^{3} - 36 x^{2} + 3 (- 36 x) + 3 \cdot 108) = 0$

Step 13.1.13.2

Multiply $- 36$ by $3$ .

$(x - 3) (x^{4} + 3 x^{3} + 9 x^{2} + 3 x^{3} - 36 x^{2} - 108 x + 3 \cdot 108) = 0$

Step 13.1.13.3

Multiply $3$ by $108$ .

$(x - 3) (x^{4} + 3 x^{3} + 9 x^{2} + 3 x^{3} - 36 x^{2} - 108 x + 324) = 0$

Step 13.1.14

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

$(x - 3) (x^{4} + 6 x^{3} + 9 x^{2} - 36 x^{2} - 108 x + 324) = 0$

Step 13.1.15

Subtract $36 x^{2}$ from $9 x^{2}$ .

$(x - 3) (x^{4} + 6 x^{3} - 27 x^{2} - 108 x + 324) = 0$

Step 13.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x^{4} + 6 x^{3} - 27 x^{2} - 108 x + 324 = 0$

Step 13.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 13.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 13.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 13.4

Set $x^{4} + 6 x^{3} - 27 x^{2} - 108 x + 324$ equal to $0$ and solve for $x$ .

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Step 13.4.1

Set $x^{4} + 6 x^{3} - 27 x^{2} - 108 x + 324$ equal to $0$ .

$x^{4} + 6 x^{3} - 27 x^{2} - 108 x + 324 = 0$

Step 13.4.2

Solve $x^{4} + 6 x^{3} - 27 x^{2} - 108 x + 324 = 0$ for $x$ .

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Step 13.4.2.1

Factor the left side of the equation.

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Step 13.4.2.1.1

Regroup terms.

$x^{4} + 6 x^{3} - 27 x^{2} - 108 x + 324 = 0$

Step 13.4.2.1.2

Factor $x^{3}$ out of $x^{4} + 6 x^{3}$ .

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Step 13.4.2.1.2.1

Factor $x^{3}$ out of $x^{4}$ .

$x^{3} x + 6 x^{3} - 27 x^{2} - 108 x + 324 = 0$

Step 13.4.2.1.2.2

Factor $x^{3}$ out of $6 x^{3}$ .

$x^{3} x + x^{3} \cdot 6 - 27 x^{2} - 108 x + 324 = 0$

Step 13.4.2.1.2.3

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot 6$ .

$x^{3} (x + 6) - 27 x^{2} - 108 x + 324 = 0$

Step 13.4.2.1.3

Factor $27$ out of $- 27 x^{2} - 108 x + 324$ .

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Step 13.4.2.1.3.1

Factor $27$ out of $- 27 x^{2}$ .

$x^{3} (x + 6) + 27 (- x^{2}) - 108 x + 324 = 0$

Step 13.4.2.1.3.2

Factor $27$ out of $- 108 x$ .

$x^{3} (x + 6) + 27 (- x^{2}) + 27 (- 4 x) + 324 = 0$

Step 13.4.2.1.3.3

Factor $27$ out of $324$ .

$x^{3} (x + 6) + 27 (- x^{2}) + 27 (- 4 x) + 27 (12) = 0$

Step 13.4.2.1.3.4

Factor $27$ out of $27 (- x^{2}) + 27 (- 4 x)$ .

$x^{3} (x + 6) + 27 (- x^{2} - 4 x) + 27 (12) = 0$

Step 13.4.2.1.3.5

Factor $27$ out of $27 (- x^{2} - 4 x) + 27 (12)$ .

$x^{3} (x + 6) + 27 (- x^{2} - 4 x + 12) = 0$

Step 13.4.2.1.4

Factor.

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Step 13.4.2.1.4.1

Factor by grouping.

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Step 13.4.2.1.4.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = - 4$ .

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Step 13.4.2.1.4.1.1.1

Factor $- 4$ out of $- 4 x$ .

$x^{3} (x + 6) + 27 (- x^{2} - 4 x + 12) = 0$

Step 13.4.2.1.4.1.1.2

Rewrite $- 4$ as $2$ plus $- 6$

$x^{3} (x + 6) + 27 (- x^{2} + (2 - 6) x + 12) = 0$

Step 13.4.2.1.4.1.1.3

Apply the distributive property.

$x^{3} (x + 6) + 27 (- x^{2} + 2 x - 6 x + 12) = 0$

Step 13.4.2.1.4.1.2

Factor out the greatest common factor from each group.

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Step 13.4.2.1.4.1.2.1

Group the first two terms and the last two terms.

$x^{3} (x + 6) + 27 ((- x^{2} + 2 x) - 6 x + 12) = 0$

Step 13.4.2.1.4.1.2.2

Factor out the greatest common factor (GCF) from each group.

$x^{3} (x + 6) + 27 (x (- x + 2) + 6 (- x + 2)) = 0$

Step 13.4.2.1.4.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$x^{3} (x + 6) + 27 ((- x + 2) (x + 6)) = 0$

Step 13.4.2.1.4.2

Remove unnecessary parentheses.

$x^{3} (x + 6) + 27 (- x + 2) (x + 6) = 0$

Step 13.4.2.1.5

Factor $x + 6$ out of $x^{3} (x + 6) + 27 (- x + 2) (x + 6)$ .

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Step 13.4.2.1.5.1

Factor $x + 6$ out of $x^{3} (x + 6)$ .

$(x + 6) x^{3} + 27 (- x + 2) (x + 6) = 0$

Step 13.4.2.1.5.2

Factor $x + 6$ out of $27 (- x + 2) (x + 6)$ .

$(x + 6) x^{3} + (x + 6) (27 (- x + 2)) = 0$

Step 13.4.2.1.5.3

Factor $x + 6$ out of $(x + 6) x^{3} + (x + 6) (27 (- x + 2))$ .

$(x + 6) (x^{3} + 27 (- x + 2)) = 0$

Step 13.4.2.1.6

Apply the distributive property.

$(x + 6) (x^{3} + 27 (- x) + 27 \cdot 2) = 0$

Step 13.4.2.1.7

Multiply $- 1$ by $27$ .

$(x + 6) (x^{3} - 27 x + 27 \cdot 2) = 0$

Step 13.4.2.1.8

Multiply $27$ by $2$ .

$(x + 6) (x^{3} - 27 x + 54) = 0$

Step 13.4.2.1.9

Factor.

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Step 13.4.2.1.9.1

Rewrite $x^{3} - 27 x + 54$ in a factored form.

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Step 13.4.2.1.9.1.1

Factor $x^{3} - 27 x + 54$ using the rational roots test.

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Step 13.4.2.1.9.1.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 54, \pm 2, \pm 27, \pm 3, \pm 18, \pm 6, \pm 9$

$q = \pm 1$

Step 13.4.2.1.9.1.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 54, \pm 2, \pm 27, \pm 3, \pm 18, \pm 6, \pm 9$

Step 13.4.2.1.9.1.1.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 13.4.2.1.9.1.1.3.1

Substitute $3$ into the polynomial.

$3^{3} - 27 \cdot 3 + 54$

Step 13.4.2.1.9.1.1.3.2

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

$27 - 27 \cdot 3 + 54$

Step 13.4.2.1.9.1.1.3.3

Multiply $- 27$ by $3$ .

$27 - 81 + 54$

Step 13.4.2.1.9.1.1.3.4

Subtract $81$ from $27$ .

$- 54 + 54$

Step 13.4.2.1.9.1.1.3.5

Add $- 54$ and $54$ .

$0$

Step 13.4.2.1.9.1.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 27 x + 54}{x - 3}$

Step 13.4.2.1.9.1.1.5

Divide $x^{3} - 27 x + 54$ by $x - 3$ .

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Step 13.4.2.1.9.1.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$3$

$x^{3}$

$0 x^{2}$

$27 x$

$54$

Step 13.4.2.1.9.1.1.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$

Step 13.4.2.1.9.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			+	$x^{3}$	-	$3 x^{2}$

Step 13.4.2.1.9.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 3 x^{2}$

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$

Step 13.4.2.1.9.1.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$

Step 13.4.2.1.9.1.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$

Step 13.4.2.1.9.1.1.5.7

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$

Step 13.4.2.1.9.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					+	$3 x^{2}$	-	$9 x$

Step 13.4.2.1.9.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} - 9 x$

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$

Step 13.4.2.1.9.1.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$

Step 13.4.2.1.9.1.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$3 x$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$

Step 13.4.2.1.9.1.1.5.12

Divide the highest order term in the dividend $- 18 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$	-	$18$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$

Step 13.4.2.1.9.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	-	$18$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$
							-	$18 x$	+	$54$

Step 13.4.2.1.9.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x + 54$

				$x^{2}$	+	$3 x$	-	$18$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$
							+	$18 x$	-	$54$

Step 13.4.2.1.9.1.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$3 x$	-	$18$
$x$	-	$3$		$x^{3}$	+	$0 x^{2}$	-	$27 x$	+	$54$
			-	$x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	-	$27 x$
					-	$3 x^{2}$	+	$9 x$
							-	$18 x$	+	$54$
							+	$18 x$	-	$54$
										$0$

Step 13.4.2.1.9.1.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 3 x - 18$

Step 13.4.2.1.9.1.1.6

Write $x^{3} - 27 x + 54$ as a set of factors.

$(x + 6) ((x - 3) (x^{2} + 3 x - 18)) = 0$

Step 13.4.2.1.9.1.2

Factor $x^{2} + 3 x - 18$ using the AC method.

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Step 13.4.2.1.9.1.2.1

Factor $x^{2} + 3 x - 18$ using the AC method.

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Step 13.4.2.1.9.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $3$ .

$- 3, 6$

Step 13.4.2.1.9.1.2.1.2

Write the factored form using these integers.

$(x + 6) ((x - 3) ((x - 3) (x + 6))) = 0$

Step 13.4.2.1.9.1.2.2

Remove unnecessary parentheses.

$(x + 6) ((x - 3) (x - 3) (x + 6)) = 0$

Step 13.4.2.1.9.1.3

Combine like factors.

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Step 13.4.2.1.9.1.3.1

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

$(x + 6) ((x - 3) (x - 3) (x + 6)) = 0$

Step 13.4.2.1.9.1.3.2

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

$(x + 6) ((x - 3) (x - 3) (x + 6)) = 0$

Step 13.4.2.1.9.1.3.3

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

$(x + 6) ({(x - 3)}^{1 + 1} (x + 6)) = 0$

Step 13.4.2.1.9.1.3.4

Add $1$ and $1$ .

$(x + 6) ({(x - 3)}^{2} (x + 6)) = 0$

Step 13.4.2.1.9.2

Remove unnecessary parentheses.

$(x + 6) {(x - 3)}^{2} (x + 6) = 0$

Step 13.4.2.1.10

Combine exponents.

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Step 13.4.2.1.10.1

Raise $x + 6$ to the power of $1$ .

$(x + 6) (x + 6) {(x - 3)}^{2} = 0$

Step 13.4.2.1.10.2

Raise $x + 6$ to the power of $1$ .

$(x + 6) (x + 6) {(x - 3)}^{2} = 0$

Step 13.4.2.1.10.3

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

${(x + 6)}^{1 + 1} {(x - 3)}^{2} = 0$

Step 13.4.2.1.10.4

Add $1$ and $1$ .

${(x + 6)}^{2} {(x - 3)}^{2} = 0$

Step 13.4.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 6)}^{2} = 0$

${(x - 3)}^{2} = 0$

Step 13.4.2.3

Set ${(x + 6)}^{2}$ equal to $0$ and solve for $x$ .

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Step 13.4.2.3.1

Set ${(x + 6)}^{2}$ equal to $0$ .

${(x + 6)}^{2} = 0$

Step 13.4.2.3.2

Solve ${(x + 6)}^{2} = 0$ for $x$ .

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Step 13.4.2.3.2.1

Set the $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 13.4.2.3.2.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 13.4.2.4

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 13.4.2.4.1

Set ${(x - 3)}^{2}$ equal to $0$ .

${(x - 3)}^{2} = 0$

Step 13.4.2.4.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 13.4.2.4.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 13.4.2.4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 13.4.2.5

The final solution is all the values that make ${(x + 6)}^{2} {(x - 3)}^{2} = 0$ true.

$x = - 6, 3$

Step 13.5

The final solution is all the values that make $(x - 3) (x^{4} + 6 x^{3} - 27 x^{2} - 108 x + 324) = 0$ true.

$x = 3, - 6$

Step 14

The polynomial can be written as a set of linear factors.

$(x - 3) (x + 6)$

Step 15

These are the roots (zeros) of the polynomial $x^{6} - 54 x^{4} + 108 x^{3} + 729 x^{2} - 2916 x + 2916$ .

$x = 3, - 6$

Step 16