Precalculus Examples

$f (x) = a (x - 6) (x - i) (x + i) (x + (9 + i)) (x + (9 - i))$

Step 1

Simplify by multiplying through.

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Step 1.1

Apply the distributive property.

$f (x) = (a x + a \cdot - 6) (x - i) (x + i) (x + 9 + i) (x + 9 - i)$

Step 1.2

Move $- 6$ to the left of $a$ .

$f (x) = (a x - 6 \cdot a) (x - i) (x + i) (x + 9 + i) (x + 9 - i)$

Step 2

Expand $(a x - 6 a) (x - i)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$f (x) = (a x (x - i) - 6 a (x - i)) (x + i) (x + 9 + i) (x + 9 - i)$

Step 2.2

Apply the distributive property.

$f (x) = (a x \cdot x + a x (- i) - 6 a (x - i)) (x + i) (x + 9 + i) (x + 9 - i)$

Step 2.3

Apply the distributive property.

$f (x) = (a x \cdot x + a x (- i) - 6 a x - 6 a (- i)) (x + i) (x + 9 + i) (x + 9 - i)$

Step 3

Simplify each term.

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Step 3.1

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

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Step 3.1.1

Move $x$ .

$f (x) = (a (x \cdot x) + a x (- i) - 6 a x - 6 a (- i)) (x + i) (x + 9 + i) (x + 9 - i)$

Step 3.1.2

Multiply $x$ by $x$ .

$f (x) = (a x^{2} + a x (- i) - 6 a x - 6 a (- i)) (x + i) (x + 9 + i) (x + 9 - i)$

Step 3.2

Multiply $- 1$ by $- 6$ .

$f (x) = (a x^{2} + a x (- i) - 6 a x + 6 a i) (x + i) (x + 9 + i) (x + 9 - i)$

Step 4

Expand $(a x^{2} + a x (- i) - 6 a x + 6 a i) (x + i)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = (a x^{2} x + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x - 6 a x i + 6 a i x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5

Simplify terms.

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Step 5.1

Combine the opposite terms in $a x^{2} x + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x - 6 a x i + 6 a i x + 6 a i i$ .

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Step 5.1.1

Reorder the factors in the terms $- 6 a x i$ and $6 a i x$ .

$f (x) = (a x^{2} x + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x - 6 i a x + 6 i a x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.1.2

Add $- 6 i a x$ and $6 i a x$ .

$f (x) = (a x^{2} x + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x + 0 + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.1.3

Add $a x^{2} x + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x$ and $0$ .

$f (x) = (a x^{2} x + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2

Simplify each term.

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Step 5.2.1

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

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Step 5.2.1.1

Move $x$ .

$f (x) = (a (x \cdot x^{2}) + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.1.2

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

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Step 5.2.1.2.1

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

$f (x) = (a (x \cdot x^{2}) + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.1.2.2

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

$f (x) = (a x^{1 + 2} + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.1.3

Add $1$ and $2$ .

$f (x) = (a x^{3} + a x^{2} i + a x (- i) x + a x (- i) i - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.2

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

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Step 5.2.2.1

Move $x$ .

$f (x) = (a x^{3} + a x^{2} i + a (x \cdot x) (- i) + a x (- i) i - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.2.2

Multiply $x$ by $x$ .

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x (- i) i - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.3

Multiply $a x (- i) i$ .

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Step 5.2.3.1

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

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x (- (i i)) - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.3.2

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

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x (- (i i)) - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.3.3

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

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

Step 5.2.3.4

Add $1$ and $1$ .

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x (- i^{2}) - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.4

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

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

Step 5.2.5

Multiply $- 1$ by $- 1$ .

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

Step 5.2.6

Multiply $a$ by $1$ .

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x - 6 a x \cdot x + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.7

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

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Step 5.2.7.1

Move $x$ .

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x - 6 a (x \cdot x) + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.7.2

Multiply $x$ by $x$ .

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x - 6 a x^{2} + 6 a i i) (x + 9 + i) (x + 9 - i)$

Step 5.2.8

Multiply $6 a i i$ .

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Step 5.2.8.1

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

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x - 6 a x^{2} + 6 a (i i)) (x + 9 + i) (x + 9 - i)$

Step 5.2.8.2

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

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x - 6 a x^{2} + 6 a (i i)) (x + 9 + i) (x + 9 - i)$

Step 5.2.8.3

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

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x - 6 a x^{2} + 6 a i^{1 + 1}) (x + 9 + i) (x + 9 - i)$

Step 5.2.8.4

Add $1$ and $1$ .

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x - 6 a x^{2} + 6 a i^{2}) (x + 9 + i) (x + 9 - i)$

Step 5.2.9

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

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

Step 5.2.10

Multiply $- 1$ by $6$ .

$f (x) = (a x^{3} + a x^{2} i + a x^{2} (- i) + a x - 6 a x^{2} - 6 a) (x + 9 + i) (x + 9 - i)$

Step 5.3

Combine the opposite terms in $a x^{3} + a x^{2} i + a x^{2} (- i) + a x - 6 a x^{2} - 6 a$ .

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Step 5.3.1

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

$f (x) = (a x^{3} + i x^{2} a - i x^{2} a + a x - 6 a x^{2} - 6 a) (x + 9 + i) (x + 9 - i)$

Step 5.3.2

Subtract $i x^{2} a$ from $i x^{2} a$ .

$f (x) = (a x^{3} + 0 + a x - 6 a x^{2} - 6 a) (x + 9 + i) (x + 9 - i)$

Step 5.3.3

Add $a x^{3}$ and $0$ .

$f (x) = (a x^{3} + a x - 6 a x^{2} - 6 a) (x + 9 + i) (x + 9 - i)$

Step 6

Expand $(a x^{3} + a x - 6 a x^{2} - 6 a) (x + 9 + i)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = (a x^{3} x + a x^{3} \cdot 9 + a x^{3} i + a x \cdot x + a x \cdot 9 + a x i - 6 a x^{2} x - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

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Step 7.1.1.1

Move $x$ .

$f (x) = (a (x \cdot x^{3}) + a x^{3} \cdot 9 + a x^{3} i + a x \cdot x + a x \cdot 9 + a x i - 6 a x^{2} x - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.1.2

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

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Step 7.1.1.2.1

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

$f (x) = (a (x \cdot x^{3}) + a x^{3} \cdot 9 + a x^{3} i + a x \cdot x + a x \cdot 9 + a x i - 6 a x^{2} x - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.1.2.2

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

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

Step 7.1.1.3

Add $1$ and $3$ .

$f (x) = (a x^{4} + a x^{3} \cdot 9 + a x^{3} i + a x \cdot x + a x \cdot 9 + a x i - 6 a x^{2} x - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.2

Move $9$ to the left of $a x^{3}$ .

$f (x) = (a x^{4} + 9 \cdot (a x^{3}) + a x^{3} i + a x \cdot x + a x \cdot 9 + a x i - 6 a x^{2} x - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.3

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

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Step 7.1.3.1

Move $x$ .

$f (x) = (a x^{4} + 9 a x^{3} + a x^{3} i + a (x \cdot x) + a x \cdot 9 + a x i - 6 a x^{2} x - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.3.2

Multiply $x$ by $x$ .

$f (x) = (a x^{4} + 9 a x^{3} + a x^{3} i + a x^{2} + a x \cdot 9 + a x i - 6 a x^{2} x - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.4

Move $9$ to the left of $a x$ .

$f (x) = (a x^{4} + 9 a x^{3} + a x^{3} i + a x^{2} + 9 \cdot (a x) + a x i - 6 a x^{2} x - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.5

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

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Step 7.1.5.1

Move $x$ .

$f (x) = (a x^{4} + 9 a x^{3} + a x^{3} i + a x^{2} + 9 a x + a x i - 6 a (x \cdot x^{2}) - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.5.2

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

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Step 7.1.5.2.1

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

$f (x) = (a x^{4} + 9 a x^{3} + a x^{3} i + a x^{2} + 9 a x + a x i - 6 a (x \cdot x^{2}) - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.5.2.2

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

$f (x) = (a x^{4} + 9 a x^{3} + a x^{3} i + a x^{2} + 9 a x + a x i - 6 a x^{1 + 2} - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.5.3

Add $1$ and $2$ .

$f (x) = (a x^{4} + 9 a x^{3} + a x^{3} i + a x^{2} + 9 a x + a x i - 6 a x^{3} - 6 a x^{2} \cdot 9 - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.6

Multiply $9$ by $- 6$ .

$f (x) = (a x^{4} + 9 a x^{3} + a x^{3} i + a x^{2} + 9 a x + a x i - 6 a x^{3} - 54 a x^{2} - 6 a x^{2} i - 6 a x - 6 a \cdot 9 - 6 a i) (x + 9 - i)$

Step 7.1.7

Multiply $9$ by $- 6$ .

$f (x) = (a x^{4} + 9 a x^{3} + a x^{3} i + a x^{2} + 9 a x + a x i - 6 a x^{3} - 54 a x^{2} - 6 a x^{2} i - 6 a x - 54 a - 6 a i) (x + 9 - i)$

Step 7.2

Simplify by adding terms.

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Step 7.2.1

Subtract $6 a x^{3}$ from $9 a x^{3}$ .

$f (x) = (a x^{4} + 3 a x^{3} + a x^{3} i + a x^{2} + 9 a x + a x i - 54 a x^{2} - 6 a x^{2} i - 6 a x - 54 a - 6 a i) (x + 9 - i)$

Step 7.2.2

Subtract $54 a x^{2}$ from $a x^{2}$ .

$f (x) = (a x^{4} + 3 a x^{3} + a x^{3} i - 53 a x^{2} + 9 a x + a x i - 6 a x^{2} i - 6 a x - 54 a - 6 a i) (x + 9 - i)$

Step 7.2.3

Subtract $6 a x$ from $9 a x$ .

$f (x) = (a x^{4} + 3 a x^{3} + a x^{3} i - 53 a x^{2} + 3 a x + a x i - 6 a x^{2} i - 54 a - 6 a i) (x + 9 - i)$

Step 8

Expand $(a x^{4} + 3 a x^{3} + a x^{3} i - 53 a x^{2} + 3 a x + a x i - 6 a x^{2} i - 54 a - 6 a i) (x + 9 - i)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = a x^{4} x + a x^{4} \cdot 9 + a x^{4} (- i) + 3 a x^{3} x + 3 a x^{3} \cdot 9 + 3 a x^{3} (- i) + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9

Simplify terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

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Step 9.1.1.1

Move $x$ .

$f (x) = a (x \cdot x^{4}) + a x^{4} \cdot 9 + a x^{4} (- i) + 3 a x^{3} x + 3 a x^{3} \cdot 9 + 3 a x^{3} (- i) + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.1.2

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

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Step 9.1.1.2.1

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

Step 9.1.1.2.2

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

$f (x) = a x^{1 + 4} + a x^{4} \cdot 9 + a x^{4} (- i) + 3 a x^{3} x + 3 a x^{3} \cdot 9 + 3 a x^{3} (- i) + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.1.3

Add $1$ and $4$ .

$f (x) = a x^{5} + a x^{4} \cdot 9 + a x^{4} (- i) + 3 a x^{3} x + 3 a x^{3} \cdot 9 + 3 a x^{3} (- i) + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.2

Move $9$ to the left of $a x^{4}$ .

$f (x) = a x^{5} + 9 \cdot (a x^{4}) + a x^{4} (- i) + 3 a x^{3} x + 3 a x^{3} \cdot 9 + 3 a x^{3} (- i) + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.3

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

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Step 9.1.3.1

Move $x$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a (x \cdot x^{3}) + 3 a x^{3} \cdot 9 + 3 a x^{3} (- i) + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.3.2

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

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Step 9.1.3.2.1

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

Step 9.1.3.2.2

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{1 + 3} + 3 a x^{3} \cdot 9 + 3 a x^{3} (- i) + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.3.3

Add $1$ and $3$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 3 a x^{3} \cdot 9 + 3 a x^{3} (- i) + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.4

Multiply $9$ by $3$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} + 3 a x^{3} (- i) + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.5

Multiply $- 1$ by $3$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{3} i x + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.6

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

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Step 9.1.6.1

Move $x$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a (x \cdot x^{3}) i + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.6.2

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

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Step 9.1.6.2.1

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

Step 9.1.6.2.2

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{1 + 3} i + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.6.3

Add $1$ and $3$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + a x^{3} i \cdot 9 + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.7

Move $9$ to the left of $a x^{3} i$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 \cdot (a x^{3} i) + a x^{3} i (- i) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.8

Multiply $a x^{3} i (- i)$ .

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Step 9.1.8.1

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} (- (i i)) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.8.2

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

Step 9.1.8.3

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} (- i^{1 + 1}) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.8.4

Add $1$ and $1$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} (- i^{2}) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.9

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} (1) - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.10

Multiply $- 1$ by $- 1$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} \cdot 1 - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.11

Multiply $a$ by $1$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{2} x - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.12

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

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Step 9.1.12.1

Move $x$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a (x \cdot x^{2}) - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.12.2

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

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Step 9.1.12.2.1

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

Step 9.1.12.2.2

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{1 + 2} - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.12.3

Add $1$ and $2$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 53 a x^{2} \cdot 9 - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.13

Multiply $9$ by $- 53$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} - 53 a x^{2} (- i) + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.14

Multiply $- 1$ by $- 53$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x \cdot x + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.15

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

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Step 9.1.15.1

Move $x$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a (x \cdot x) + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.15.2

Multiply $x$ by $x$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 3 a x \cdot 9 + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.16

Multiply $9$ by $3$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x + 3 a x (- i) + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.17

Multiply $- 1$ by $3$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x i x + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.18

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

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Step 9.1.18.1

Move $x$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a (x \cdot x) i + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.18.2

Multiply $x$ by $x$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + a x i \cdot 9 + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.19

Move $9$ to the left of $a x i$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 \cdot (a x i) + a x i (- i) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.20

Multiply $a x i (- i)$ .

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Step 9.1.20.1

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x (- (i i)) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.20.2

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

Step 9.1.20.3

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x (- i^{1 + 1}) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.20.4

Add $1$ and $1$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x (- i^{2}) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.21

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x (1) - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.22

Multiply $- 1$ by $- 1$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x \cdot 1 - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.23

Multiply $a$ by $1$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{2} i x - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.24

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

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Step 9.1.24.1

Move $x$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a (x \cdot x^{2}) i - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.24.2

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

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Step 9.1.24.2.1

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

Step 9.1.24.2.2

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{1 + 2} i - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.24.3

Add $1$ and $2$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 6 a x^{2} i \cdot 9 - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.25

Multiply $9$ by $- 6$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} i (- i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.26

Multiply $- 6 a x^{2} i (- i)$ .

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Step 9.1.26.1

Multiply $- 1$ by $- 6$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i + 6 a x^{2} i i - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.26.2

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i + 6 a x^{2} (i i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.26.3

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i + 6 a x^{2} (i i) - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.26.4

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i + 6 a x^{2} i^{1 + 1} - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.26.5

Add $1$ and $1$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i + 6 a x^{2} i^{2} - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.27

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i + 6 a x^{2} \cdot - 1 - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.28

Multiply $- 1$ by $6$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 54 a \cdot 9 - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.29

Multiply $9$ by $- 54$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 54 a (- i) - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.30

Multiply $- 1$ by $- 54$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 6 a i \cdot 9 - 6 a i (- i)$

Step 9.1.31

Multiply $9$ by $- 6$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i - 6 a i (- i)$

Step 9.1.32

Multiply $- 6 a i (- i)$ .

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Step 9.1.32.1

Multiply $- 1$ by $- 6$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i + 6 a i i$

Step 9.1.32.2

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i + 6 a (i i)$

Step 9.1.32.3

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i + 6 a (i i)$

Step 9.1.32.4

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i + 6 a i^{1 + 1}$

Step 9.1.32.5

Add $1$ and $1$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i + 6 a i^{2}$

Step 9.1.33

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

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i + 6 a \cdot - 1$

Step 9.1.34

Multiply $- 1$ by $6$ .

$f (x) = a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i - 6 a$

Step 9.2

Simplify by adding terms.

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Step 9.2.1

Combine the opposite terms in $a x^{5} + 9 a x^{4} + a x^{4} (- i) + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + a x^{4} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i - 6 a$ .

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Step 9.2.1.1

Reorder the factors in the terms $a x^{4} (- i)$ and $a x^{4} i$ .

$f (x) = a x^{5} + 9 a x^{4} - i x^{4} a + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + i x^{4} a + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i - 6 a$

Step 9.2.1.2

Add $- i x^{4} a$ and $i x^{4} a$ .

$f (x) = a x^{5} + 9 a x^{4} + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + 0 + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i - 6 a$

Step 9.2.1.3

Add $a x^{5} + 9 a x^{4} + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i$ and $0$ .

$f (x) = a x^{5} + 9 a x^{4} + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a + 54 a i - 6 a i x - 54 a i - 6 a$

Step 9.2.1.4

Subtract $54 a i$ from $54 a i$ .

Step 9.2.1.5

Add $a x^{5} + 9 a x^{4} + 3 a x^{4} + 27 a x^{3} - 3 a x^{3} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 6 a i x$ and $0$ .

Step 9.2.2

Add $9 a x^{4}$ and $3 a x^{4}$ .

$f (x) = a x^{5} + 12 a x^{4} + 27 a x^{3} - 3 a x^{3} i + 9 a x^{3} i + a x^{3} - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.3

Add $27 a x^{3}$ and $a x^{3}$ .

$f (x) = a x^{5} + 12 a x^{4} + 28 a x^{3} - 3 a x^{3} i + 9 a x^{3} i - 53 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.4

Subtract $53 a x^{3}$ from $28 a x^{3}$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 3 a x^{3} i + 9 a x^{3} i - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.5

Add $- 3 a x^{3} i$ and $9 a x^{3} i$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} + 6 a x^{3} i - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.6

Combine the opposite terms in $a x^{5} + 12 a x^{4} - 25 a x^{3} + 6 a x^{3} i - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 6 a x^{3} i - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 6 a i x - 6 a$ .

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Step 9.2.6.1

Subtract $6 a x^{3} i$ from $6 a x^{3} i$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x + 0 - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.6.2

Add $a x^{5} + 12 a x^{4} - 25 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x$ and $0$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 477 a x^{2} + 53 a x^{2} i + 3 a x^{2} + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.7

Add $- 477 a x^{2}$ and $3 a x^{2}$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 474 a x^{2} + 53 a x^{2} i + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 54 a x^{2} i - 6 a x^{2} - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.8

Subtract $6 a x^{2}$ from $- 474 a x^{2}$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} + 53 a x^{2} i + 27 a x - 3 a x i + a x^{2} i + 9 a x i + a x - 54 a x^{2} i - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.9

Add $53 a x^{2} i$ and $a x^{2} i$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} + 54 a x^{2} i + 27 a x - 3 a x i + 9 a x i + a x - 54 a x^{2} i - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.10

Combine the opposite terms in $a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} + 54 a x^{2} i + 27 a x - 3 a x i + 9 a x i + a x - 54 a x^{2} i - 54 a x - 486 a - 6 a i x - 6 a$ .

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Step 9.2.10.1

Subtract $54 a x^{2} i$ from $54 a x^{2} i$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} + 27 a x - 3 a x i + 9 a x i + a x + 0 - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.10.2

Add $a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} + 27 a x - 3 a x i + 9 a x i + a x$ and $0$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} + 27 a x - 3 a x i + 9 a x i + a x - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.11

Add $27 a x$ and $a x$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} + 28 a x - 3 a x i + 9 a x i - 54 a x - 486 a - 6 a i x - 6 a$

Step 9.2.12

Subtract $54 a x$ from $28 a x$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} - 26 a x - 3 a x i + 9 a x i - 486 a - 6 a i x - 6 a$

Step 9.2.13

Add $- 3 a x i$ and $9 a x i$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} - 26 a x + 6 a x i - 486 a - 6 a i x - 6 a$

Step 9.2.14

Combine the opposite terms in $a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} - 26 a x + 6 a x i - 486 a - 6 a i x - 6 a$ .

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Step 9.2.14.1

Reorder the factors in the terms $6 a x i$ and $- 6 a i x$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} - 26 a x + 6 i a x - 486 a - 6 i a x - 6 a$

Step 9.2.14.2

Subtract $6 i a x$ from $6 i a x$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} - 26 a x - 486 a + 0 - 6 a$

Step 9.2.14.3

Add $a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} - 26 a x - 486 a$ and $0$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} - 26 a x - 486 a - 6 a$

Step 9.2.15

Subtract $6 a$ from $- 486 a$ .

$f (x) = a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} - 26 a x - 492 a$

Step 10

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$a x^{5} \to 6$

$12 a x^{4} \to 5$

$- 25 a x^{3} \to 4$

$- 480 a x^{2} \to 3$

$- 26 a x \to 2$

$- 492 a \to 1$

Step 11

The largest exponent is the degree of the polynomial.

$6$

Step 12

The maximum number of possible roots is the degree of $a x^{5} + 12 a x^{4} - 25 a x^{3} - 480 a x^{2} - 26 a x - 492 a$ .

$6$