Precalculus Examples

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

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.3.2

Subtract $2 x$ from $- 2 x$ .

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

Step 1.4

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

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

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

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Step 1.5.1.1.1

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

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Step 1.5.1.1.1.1

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

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

Step 1.5.1.1.1.2

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

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

Step 1.5.1.1.2

Add $2$ and $1$ .

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

Step 1.5.1.2

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

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

Step 1.5.1.3

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

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Step 1.5.1.3.1

Move $x$ .

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

Step 1.5.1.3.2

Multiply $x$ by $x$ .

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

Step 1.5.1.4

Multiply $3$ by $- 4$ .

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

Step 1.5.1.5

Multiply $4$ by $3$ .

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

Step 1.5.2

Simplify by adding terms.

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Step 1.5.2.1

Subtract $4 x^{2}$ from $3 x^{2}$ .

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

Step 1.5.2.2

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

$f (x) = (x^{3} - x^{2} - 8 x + 12) {(x + 1)}^{2}$

Step 1.5.2.3

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$f (x) = (x^{3} - x^{2} - 8 x + 12) ((x + 1) (x + 1))$

Step 1.6

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

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Step 1.6.1

Apply the distributive property.

$f (x) = (x^{3} - x^{2} - 8 x + 12) (x (x + 1) + 1 (x + 1))$

Step 1.6.2

Apply the distributive property.

$f (x) = (x^{3} - x^{2} - 8 x + 12) (x \cdot x + x \cdot 1 + 1 (x + 1))$

Step 1.6.3

Apply the distributive property.

$f (x) = (x^{3} - x^{2} - 8 x + 12) (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1)$

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $x$ by $x$ .

$f (x) = (x^{3} - x^{2} - 8 x + 12) (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1)$

Step 1.7.1.2

Multiply $x$ by $1$ .

$f (x) = (x^{3} - x^{2} - 8 x + 12) (x^{2} + x + 1 x + 1 \cdot 1)$

Step 1.7.1.3

Multiply $x$ by $1$ .

$f (x) = (x^{3} - x^{2} - 8 x + 12) (x^{2} + x + x + 1 \cdot 1)$

Step 1.7.1.4

Multiply $1$ by $1$ .

$f (x) = (x^{3} - x^{2} - 8 x + 12) (x^{2} + x + x + 1)$

Step 1.7.2

Add $x$ and $x$ .

$f (x) = (x^{3} - x^{2} - 8 x + 12) (x^{2} + 2 x + 1)$

Step 1.8

Expand $(x^{3} - x^{2} - 8 x + 12) (x^{2} + 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = x^{3} x^{2} + x^{3} (2 x) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9

Simplify terms.

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

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

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Step 1.9.1.1.1

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

$f (x) = x^{3 + 2} + x^{3} (2 x) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.1.2

Add $3$ and $2$ .

$f (x) = x^{5} + x^{3} (2 x) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.2

Rewrite using the commutative property of multiplication.

$f (x) = x^{5} + 2 x^{3} x + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.3

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

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Step 1.9.1.3.1

Move $x$ .

$f (x) = x^{5} + 2 (x \cdot x^{3}) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.3.2

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

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Step 1.9.1.3.2.1

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

$f (x) = x^{5} + 2 (x \cdot x^{3}) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.3.2.2

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

$f (x) = x^{5} + 2 x^{1 + 3} + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.3.3

Add $1$ and $3$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.4

Multiply $x^{3}$ by $1$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.5

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

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Step 1.9.1.5.1

Move $x^{2}$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - (x^{2} x^{2}) - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.5.2

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

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{2 + 2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.5.3

Add $2$ and $2$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.6

Rewrite using the commutative property of multiplication.

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot (2 x^{2} x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.7

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

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Step 1.9.1.7.1

Move $x$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot (2 (x \cdot x^{2})) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.7.2

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

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Step 1.9.1.7.2.1

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

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot (2 (x \cdot x^{2})) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.7.2.2

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

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot (2 x^{1 + 2}) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.7.3

Add $1$ and $2$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot (2 x^{3}) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.8

Multiply $- 1$ by $2$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.9

Multiply $- 1$ by $1$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.10

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

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Step 1.9.1.10.1

Move $x^{2}$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 (x^{2} x) - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.10.2

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

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Step 1.9.1.10.2.1

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

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 (x^{2} x) - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.10.2.2

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

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{2 + 1} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.10.3

Add $2$ and $1$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.11

Rewrite using the commutative property of multiplication.

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 \cdot (2 x \cdot x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.12

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

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Step 1.9.1.12.1

Move $x$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 \cdot (2 (x \cdot x)) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.12.2

Multiply $x$ by $x$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 \cdot (2 x^{2}) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.13

Multiply $- 8$ by $2$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.14

Multiply $- 8$ by $1$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Step 1.9.1.15

Multiply $2$ by $12$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12 \cdot 1$

Step 1.9.1.16

Multiply $12$ by $1$ .

$f (x) = x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Step 1.9.2

Simplify by adding terms.

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Step 1.9.2.1

Subtract $x^{4}$ from $2 x^{4}$ .

$f (x) = x^{5} + x^{4} + x^{3} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Step 1.9.2.2

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

$f (x) = x^{5} + x^{4} - x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Step 1.9.2.3

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

$f (x) = x^{5} + x^{4} - 9 x^{3} - x^{2} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Step 1.9.2.4

Subtract $16 x^{2}$ from $- x^{2}$ .

$f (x) = x^{5} + x^{4} - 9 x^{3} - 17 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Step 1.9.2.5

Add $- 17 x^{2}$ and $12 x^{2}$ .

$f (x) = x^{5} + x^{4} - 9 x^{3} - 5 x^{2} - 8 x + 24 x + 12$

Step 1.9.2.6

Add $- 8 x$ and $24 x$ .

$f (x) = x^{5} + x^{4} - 9 x^{3} - 5 x^{2} + 16 x + 12$

Step 2

Find $f (- x)$ .

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Step 2.1

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = {(- x)}^{5} + {(- x)}^{4} - 9 {(- x)}^{3} - 5 {(- x)}^{2} + 16 (- x) + 12$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{5} x^{5} + {(- x)}^{4} - 9 {(- x)}^{3} - 5 {(- x)}^{2} + 16 (- x) + 12$

Step 2.2.2

Raise $- 1$ to the power of $5$ .

$f (- x) = - x^{5} + {(- x)}^{4} - 9 {(- x)}^{3} - 5 {(- x)}^{2} + 16 (- x) + 12$

Step 2.2.3

Apply the product rule to $- x$ .

$f (- x) = - x^{5} + {(- 1)}^{4} x^{4} - 9 {(- x)}^{3} - 5 {(- x)}^{2} + 16 (- x) + 12$

Step 2.2.4

Raise $- 1$ to the power of $4$ .

$f (- x) = - x^{5} + 1 x^{4} - 9 {(- x)}^{3} - 5 {(- x)}^{2} + 16 (- x) + 12$

Step 2.2.5

Multiply $x^{4}$ by $1$ .

$f (- x) = - x^{5} + x^{4} - 9 {(- x)}^{3} - 5 {(- x)}^{2} + 16 (- x) + 12$

Step 2.2.6

Apply the product rule to $- x$ .

$f (- x) = - x^{5} + x^{4} - 9 ({(- 1)}^{3} x^{3}) - 5 {(- x)}^{2} + 16 (- x) + 12$

Step 2.2.7

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

$f (- x) = - x^{5} + x^{4} - 9 (- x^{3}) - 5 {(- x)}^{2} + 16 (- x) + 12$

Step 2.2.8

Multiply $- 1$ by $- 9$ .

$f (- x) = - x^{5} + x^{4} + 9 x^{3} - 5 {(- x)}^{2} + 16 (- x) + 12$

Step 2.2.9

Apply the product rule to $- x$ .

$f (- x) = - x^{5} + x^{4} + 9 x^{3} - 5 ({(- 1)}^{2} x^{2}) + 16 (- x) + 12$

Step 2.2.10

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

$f (- x) = - x^{5} + x^{4} + 9 x^{3} - 5 (1 x^{2}) + 16 (- x) + 12$

Step 2.2.11

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

$f (- x) = - x^{5} + x^{4} + 9 x^{3} - 5 x^{2} + 16 (- x) + 12$

Step 2.2.12

Multiply $- 1$ by $16$ .

$f (- x) = - x^{5} + x^{4} + 9 x^{3} - 5 x^{2} - 16 x + 12$

Step 3

A function is even if $f (- x) = f (x)$ .

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Step 3.1

Check if $f (- x) = f (x)$ .

Step 3.2

Since $- x^{5} + x^{4} + 9 x^{3} - 5 x^{2} - 16 x + 12$ $\neq$ $x^{5} + x^{4} - 9 x^{3} - 5 x^{2} + 16 x + 12$ , the function is not even.

The function is not even

Step 4

A function is odd if $f (- x) = - f (x)$ .

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Step 4.1

Find $- f (x)$ .

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Step 4.1.1

Multiply $x^{5} + x^{4} - 9 x^{3} - 5 x^{2} + 16 x + 12$ by $- 1$ .

$- f (x) = - (x^{5} + x^{4} - 9 x^{3} - 5 x^{2} + 16 x + 12)$

Step 4.1.2

Apply the distributive property.

$- f (x) = - x^{5} - x^{4} - (- 9 x^{3}) - (- 5 x^{2}) - (16 x) - 1 \cdot 12$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Multiply $- 9$ by $- 1$ .

$- f (x) = - x^{5} - x^{4} + 9 x^{3} - (- 5 x^{2}) - (16 x) - 1 \cdot 12$

Step 4.1.3.2

Multiply $- 5$ by $- 1$ .

$- f (x) = - x^{5} - x^{4} + 9 x^{3} + 5 x^{2} - (16 x) - 1 \cdot 12$

Step 4.1.3.3

Multiply $16$ by $- 1$ .

$- f (x) = - x^{5} - x^{4} + 9 x^{3} + 5 x^{2} - 16 x - 1 \cdot 12$

Step 4.1.3.4

Multiply $- 1$ by $12$ .

$- f (x) = - x^{5} - x^{4} + 9 x^{3} + 5 x^{2} - 16 x - 12$

Step 4.2

Since $- x^{5} + x^{4} + 9 x^{3} - 5 x^{2} - 16 x + 12$ $\neq$ $- x^{5} - x^{4} + 9 x^{3} + 5 x^{2} - 16 x - 12$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6