Algebra Examples

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

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Apply the product rule to $- x$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Apply the product rule to $- x$ .

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

Step 1.2.1.4

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

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

Step 1.2.1.5

Multiply $x^{2}$ by $1$ .

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

Step 1.2.1.6

Multiply $3$ by $1$ .

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

Step 1.2.1.7

Multiply $- 1$ by $3$ .

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

Step 1.2.1.8

One to any power is one.

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

Step 1.2.1.9

Multiply $- 3$ by $1$ .

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

Step 1.2.1.10

One to any power is one.

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ .

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

Step 1.4.1.2

Move $2$ to the left of $x$ .

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

Step 1.4.1.3

Multiply $2$ by $2$ .

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

Step 1.4.2

Add $2 x$ and $2 x$ .

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

Step 1.5

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

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

Step 1.6

Simplify terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

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

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Step 1.6.1.1.1

Move $x^{2}$ .

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

Step 1.6.1.1.2

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

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

Step 1.6.1.1.3

Add $2$ and $3$ .

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

Step 1.6.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.3

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

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Step 1.6.1.3.1

Move $x$ .

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

Step 1.6.1.3.2

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

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Step 1.6.1.3.2.1

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

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

Step 1.6.1.3.2.2

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

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

Step 1.6.1.3.3

Add $1$ and $3$ .

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

Step 1.6.1.4

Multiply $- 1$ by $4$ .

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

Step 1.6.1.5

Multiply $4$ by $- 1$ .

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

Step 1.6.1.6

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

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Step 1.6.1.6.1

Move $x^{2}$ .

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

Step 1.6.1.6.2

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

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

Step 1.6.1.6.3

Add $2$ and $2$ .

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

Step 1.6.1.7

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.8

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

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Step 1.6.1.8.1

Move $x$ .

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

Step 1.6.1.8.2

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

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Step 1.6.1.8.2.1

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

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

Step 1.6.1.8.2.2

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

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

Step 1.6.1.8.3

Add $1$ and $2$ .

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

Step 1.6.1.9

Multiply $3$ by $4$ .

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

Step 1.6.1.10

Multiply $4$ by $3$ .

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

Step 1.6.1.11

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

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Step 1.6.1.11.1

Move $x^{2}$ .

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

Step 1.6.1.11.2

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

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Step 1.6.1.11.2.1

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

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

Step 1.6.1.11.2.2

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

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

Step 1.6.1.11.3

Add $2$ and $1$ .

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

Step 1.6.1.12

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.13

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

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Step 1.6.1.13.1

Move $x$ .

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

Step 1.6.1.13.2

Multiply $x$ by $x$ .

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

Step 1.6.1.14

Multiply $- 3$ by $4$ .

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

Step 1.6.1.15

Multiply $4$ by $- 3$ .

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

Step 1.6.1.16

Multiply $x^{2}$ by $1$ .

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

Step 1.6.1.17

Multiply $4 x$ by $1$ .

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

Step 1.6.1.18

Multiply $4$ by $1$ .

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

Step 1.6.2

Simplify by adding terms.

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Step 1.6.2.1

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

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Step 1.6.2.1.1

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

$(- x^{5} - 4 x^{4} - 4 x^{3} + 3 x^{4} + 12 x^{3} - 3 x^{3} + 0 - 12 x + x^{2} + 4 x + 4) (x - 3)$

Step 1.6.2.1.2

Add $- x^{5} - 4 x^{4} - 4 x^{3} + 3 x^{4} + 12 x^{3} - 3 x^{3}$ and $0$ .

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

Step 1.6.2.2

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

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

Step 1.6.2.3

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

$(- x^{5} - x^{4} + 8 x^{3} - 3 x^{3} - 12 x + x^{2} + 4 x + 4) (x - 3)$

Step 1.6.2.4

Subtract $3 x^{3}$ from $8 x^{3}$ .

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

Step 1.6.2.5

Add $- 12 x$ and $4 x$ .

$(- x^{5} - x^{4} + 5 x^{3} + x^{2} - 8 x + 4) (x - 3)$

Step 1.7

Expand $(- x^{5} - x^{4} + 5 x^{3} + x^{2} - 8 x + 4) (x - 3)$ by multiplying each term in the first expression by each term in the second expression.

$- x^{5} x - x^{5} \cdot - 3 - x^{4} x - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8

Simplify terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 1.8.1.1.1

Move $x$ .

$- (x \cdot x^{5}) - x^{5} \cdot - 3 - x^{4} x - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.1.2

Multiply $x$ by $x^{5}$ .

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Step 1.8.1.1.2.1

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

$- (x^{1} x^{5}) - x^{5} \cdot - 3 - x^{4} x - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.1.2.2

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

$- x^{1 + 5} - x^{5} \cdot - 3 - x^{4} x - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.1.3

Add $1$ and $5$ .

$- x^{6} - x^{5} \cdot - 3 - x^{4} x - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.2

Multiply $- 3$ by $- 1$ .

$- x^{6} + 3 x^{5} - x^{4} x - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.3

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

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Step 1.8.1.3.1

Move $x$ .

$- x^{6} + 3 x^{5} - (x \cdot x^{4}) - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.3.2

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

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Step 1.8.1.3.2.1

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

$- x^{6} + 3 x^{5} - (x^{1} x^{4}) - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.3.2.2

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

$- x^{6} + 3 x^{5} - x^{1 + 4} - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.3.3

Add $1$ and $4$ .

$- x^{6} + 3 x^{5} - x^{5} - x^{4} \cdot - 3 + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.4

Multiply $- 3$ by $- 1$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{3} x + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.5

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

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Step 1.8.1.5.1

Move $x$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 (x \cdot x^{3}) + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.5.2

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

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Step 1.8.1.5.2.1

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

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 (x^{1} x^{3}) + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.5.2.2

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

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{1 + 3} + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.5.3

Add $1$ and $3$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} + 5 x^{3} \cdot - 3 + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.6

Multiply $- 3$ by $5$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{2} x + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.7

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

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Step 1.8.1.7.1

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

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Step 1.8.1.7.1.1

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

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{2} x^{1} + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.7.1.2

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

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{2 + 1} + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.7.2

Add $2$ and $1$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{3} + x^{2} \cdot - 3 - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.8

Move $- 3$ to the left of $x^{2}$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{3} - 3 \cdot x^{2} - 8 x \cdot x - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.9

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

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Step 1.8.1.9.1

Move $x$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{3} - 3 x^{2} - 8 (x \cdot x) - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.9.2

Multiply $x$ by $x$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{3} - 3 x^{2} - 8 x^{2} - 8 x \cdot - 3 + 4 x + 4 \cdot - 3$

Step 1.8.1.10

Multiply $- 3$ by $- 8$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{3} - 3 x^{2} - 8 x^{2} + 24 x + 4 x + 4 \cdot - 3$

Step 1.8.1.11

Multiply $4$ by $- 3$ .

$- x^{6} + 3 x^{5} - x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{3} - 3 x^{2} - 8 x^{2} + 24 x + 4 x - 12$

Step 1.8.2

Simplify by adding terms.

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Step 1.8.2.1

Subtract $x^{5}$ from $3 x^{5}$ .

$- x^{6} + 2 x^{5} + 3 x^{4} + 5 x^{4} - 15 x^{3} + x^{3} - 3 x^{2} - 8 x^{2} + 24 x + 4 x - 12$

Step 1.8.2.2

Add $3 x^{4}$ and $5 x^{4}$ .

$- x^{6} + 2 x^{5} + 8 x^{4} - 15 x^{3} + x^{3} - 3 x^{2} - 8 x^{2} + 24 x + 4 x - 12$

Step 1.8.2.3

Add $- 15 x^{3}$ and $x^{3}$ .

$- x^{6} + 2 x^{5} + 8 x^{4} - 14 x^{3} - 3 x^{2} - 8 x^{2} + 24 x + 4 x - 12$

Step 1.8.2.4

Subtract $8 x^{2}$ from $- 3 x^{2}$ .

$- x^{6} + 2 x^{5} + 8 x^{4} - 14 x^{3} - 11 x^{2} + 24 x + 4 x - 12$

Step 1.8.2.5

Add $24 x$ and $4 x$ .

$- x^{6} + 2 x^{5} + 8 x^{4} - 14 x^{3} - 11 x^{2} + 28 x - 12$

Step 2

The largest exponent is the degree of the polynomial.

$6$