Algebra Examples

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

Step 1

Identify the degree of the function.

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Step 1.1

Simplify and reorder the polynomial.

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Step 1.1.1

Simplify by multiplying through.

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Multiply $3$ by $3$ .

$(3 x + 9) (x + 2) {(x - 1)}^{3}$

Step 1.1.2

Expand $(3 x + 9) (x + 2)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$(3 x (x + 2) + 9 (x + 2)) {(x - 1)}^{3}$

Step 1.1.2.2

Apply the distributive property.

$(3 x \cdot x + 3 x \cdot 2 + 9 (x + 2)) {(x - 1)}^{3}$

Step 1.1.2.3

Apply the distributive property.

$(3 x \cdot x + 3 x \cdot 2 + 9 x + 9 \cdot 2) {(x - 1)}^{3}$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

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

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Step 1.1.3.1.1.1

Move $x$ .

$(3 (x \cdot x) + 3 x \cdot 2 + 9 x + 9 \cdot 2) {(x - 1)}^{3}$

Step 1.1.3.1.1.2

Multiply $x$ by $x$ .

$(3 x^{2} + 3 x \cdot 2 + 9 x + 9 \cdot 2) {(x - 1)}^{3}$

Step 1.1.3.1.2

Multiply $2$ by $3$ .

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

Step 1.1.3.1.3

Multiply $9$ by $2$ .

$(3 x^{2} + 6 x + 9 x + 18) {(x - 1)}^{3}$

Step 1.1.3.2

Add $6 x$ and $9 x$ .

$(3 x^{2} + 15 x + 18) {(x - 1)}^{3}$

Step 1.1.4

Use the Binomial Theorem.

$(3 x^{2} + 15 x + 18) (x^{3} + 3 x^{2} \cdot - 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.1.5

Simplify each term.

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Step 1.1.5.1

Multiply $- 1$ by $3$ .

$(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.1.5.2

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

$(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x \cdot 1 + {(- 1)}^{3})$

Step 1.1.5.3

Multiply $3$ by $1$ .

$(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x + {(- 1)}^{3})$

Step 1.1.5.4

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

$(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x - 1)$

Step 1.1.6

Expand $(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{2} x^{3} + 3 x^{2} (- 3 x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7

Simplify terms.

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Step 1.1.7.1

Simplify each term.

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Step 1.1.7.1.1

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

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Step 1.1.7.1.1.1

Move $x^{3}$ .

$3 (x^{3} x^{2}) + 3 x^{2} (- 3 x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.1.2

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

$3 x^{3 + 2} + 3 x^{2} (- 3 x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.1.3

Add $3$ and $2$ .

$3 x^{5} + 3 x^{2} (- 3 x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.2

Rewrite using the commutative property of multiplication.

$3 x^{5} + 3 \cdot - 3 x^{2} x^{2} + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.3

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

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Step 1.1.7.1.3.1

Move $x^{2}$ .

$3 x^{5} + 3 \cdot - 3 (x^{2} x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.3.2

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

$3 x^{5} + 3 \cdot - 3 x^{2 + 2} + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.3.3

Add $2$ and $2$ .

$3 x^{5} + 3 \cdot - 3 x^{4} + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.4

Multiply $3$ by $- 3$ .

$3 x^{5} - 9 x^{4} + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.5

Rewrite using the commutative property of multiplication.

$3 x^{5} - 9 x^{4} + 3 \cdot 3 x^{2} x + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.6

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

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Step 1.1.7.1.6.1

Move $x$ .

$3 x^{5} - 9 x^{4} + 3 \cdot 3 (x \cdot x^{2}) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.6.2

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

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Step 1.1.7.1.6.2.1

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

$3 x^{5} - 9 x^{4} + 3 \cdot 3 (x^{1} x^{2}) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.6.2.2

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

$3 x^{5} - 9 x^{4} + 3 \cdot 3 x^{1 + 2} + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.6.3

Add $1$ and $2$ .

$3 x^{5} - 9 x^{4} + 3 \cdot 3 x^{3} + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.7

Multiply $3$ by $3$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.8

Multiply $- 1$ by $3$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.9

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

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Step 1.1.7.1.9.1

Move $x^{3}$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 (x^{3} x) + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.9.2

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

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Step 1.1.7.1.9.2.1

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

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 (x^{3} x^{1}) + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.9.2.2

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

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{3 + 1} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.9.3

Add $3$ and $1$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.10

Rewrite using the commutative property of multiplication.

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 x \cdot x^{2} + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.11

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

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Step 1.1.7.1.11.1

Move $x^{2}$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 (x^{2} x) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.11.2

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

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Step 1.1.7.1.11.2.1

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

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 (x^{2} x^{1}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.11.2.2

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

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 x^{2 + 1} + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.11.3

Add $2$ and $1$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 x^{3} + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.12

Multiply $15$ by $- 3$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.13

Rewrite using the commutative property of multiplication.

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 15 \cdot 3 x \cdot x + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.14

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

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Step 1.1.7.1.14.1

Move $x$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 15 \cdot 3 (x \cdot x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.14.2

Multiply $x$ by $x$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 15 \cdot 3 x^{2} + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.15

Multiply $15$ by $3$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.16

Multiply $- 1$ by $15$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.17

Multiply $- 3$ by $18$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 18 (3 x) + 18 \cdot - 1$

Step 1.1.7.1.18

Multiply $3$ by $18$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 54 x + 18 \cdot - 1$

Step 1.1.7.1.19

Multiply $18$ by $- 1$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 54 x - 18$

Step 1.1.7.2

Simplify by adding terms.

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Step 1.1.7.2.1

Add $- 9 x^{4}$ and $15 x^{4}$ .

$3 x^{5} + 6 x^{4} + 9 x^{3} - 3 x^{2} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 54 x - 18$

Step 1.1.7.2.2

Subtract $45 x^{3}$ from $9 x^{3}$ .

$3 x^{5} + 6 x^{4} - 36 x^{3} - 3 x^{2} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 54 x - 18$

Step 1.1.7.2.3

Add $- 36 x^{3}$ and $18 x^{3}$ .

$3 x^{5} + 6 x^{4} - 18 x^{3} - 3 x^{2} + 45 x^{2} - 15 x - 54 x^{2} + 54 x - 18$

Step 1.1.7.2.4

Add $- 3 x^{2}$ and $45 x^{2}$ .

$3 x^{5} + 6 x^{4} - 18 x^{3} + 42 x^{2} - 15 x - 54 x^{2} + 54 x - 18$

Step 1.1.7.2.5

Subtract $54 x^{2}$ from $42 x^{2}$ .

$3 x^{5} + 6 x^{4} - 18 x^{3} - 12 x^{2} - 15 x + 54 x - 18$

Step 1.1.7.2.6

Add $- 15 x$ and $54 x$ .

$3 x^{5} + 6 x^{4} - 18 x^{3} - 12 x^{2} + 39 x - 18$

Step 1.2

The largest exponent is the degree of the polynomial.

$5$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

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Step 3.1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 3.1.1

Simplify by multiplying through.

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Step 3.1.1.1

Apply the distributive property.

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

Step 3.1.1.2

Multiply $3$ by $3$ .

$(3 x + 9) (x + 2) {(x - 1)}^{3}$

Step 3.1.2

Expand $(3 x + 9) (x + 2)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$(3 x (x + 2) + 9 (x + 2)) {(x - 1)}^{3}$

Step 3.1.2.2

Apply the distributive property.

$(3 x \cdot x + 3 x \cdot 2 + 9 (x + 2)) {(x - 1)}^{3}$

Step 3.1.2.3

Apply the distributive property.

$(3 x \cdot x + 3 x \cdot 2 + 9 x + 9 \cdot 2) {(x - 1)}^{3}$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

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

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Step 3.1.3.1.1.1

Move $x$ .

$(3 (x \cdot x) + 3 x \cdot 2 + 9 x + 9 \cdot 2) {(x - 1)}^{3}$

Step 3.1.3.1.1.2

Multiply $x$ by $x$ .

$(3 x^{2} + 3 x \cdot 2 + 9 x + 9 \cdot 2) {(x - 1)}^{3}$

Step 3.1.3.1.2

Multiply $2$ by $3$ .

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

Step 3.1.3.1.3

Multiply $9$ by $2$ .

$(3 x^{2} + 6 x + 9 x + 18) {(x - 1)}^{3}$

Step 3.1.3.2

Add $6 x$ and $9 x$ .

$(3 x^{2} + 15 x + 18) {(x - 1)}^{3}$

Step 3.1.4

Use the Binomial Theorem.

$(3 x^{2} + 15 x + 18) (x^{3} + 3 x^{2} \cdot - 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3})$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

Multiply $- 1$ by $3$ .

$(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3})$

Step 3.1.5.2

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

$(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x \cdot 1 + {(- 1)}^{3})$

Step 3.1.5.3

Multiply $3$ by $1$ .

$(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x + {(- 1)}^{3})$

Step 3.1.5.4

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

$(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x - 1)$

Step 3.1.6

Expand $(3 x^{2} + 15 x + 18) (x^{3} - 3 x^{2} + 3 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{2} x^{3} + 3 x^{2} (- 3 x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7

Simplify terms.

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Step 3.1.7.1

Simplify each term.

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Step 3.1.7.1.1

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

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Step 3.1.7.1.1.1

Move $x^{3}$ .

$3 (x^{3} x^{2}) + 3 x^{2} (- 3 x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.1.2

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

$3 x^{3 + 2} + 3 x^{2} (- 3 x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.1.3

Add $3$ and $2$ .

$3 x^{5} + 3 x^{2} (- 3 x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.2

Rewrite using the commutative property of multiplication.

$3 x^{5} + 3 \cdot - 3 x^{2} x^{2} + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.3

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

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Step 3.1.7.1.3.1

Move $x^{2}$ .

$3 x^{5} + 3 \cdot - 3 (x^{2} x^{2}) + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.3.2

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

$3 x^{5} + 3 \cdot - 3 x^{2 + 2} + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.3.3

Add $2$ and $2$ .

$3 x^{5} + 3 \cdot - 3 x^{4} + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.4

Multiply $3$ by $- 3$ .

$3 x^{5} - 9 x^{4} + 3 x^{2} (3 x) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.5

Rewrite using the commutative property of multiplication.

$3 x^{5} - 9 x^{4} + 3 \cdot 3 x^{2} x + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.6

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

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Step 3.1.7.1.6.1

Move $x$ .

$3 x^{5} - 9 x^{4} + 3 \cdot 3 (x \cdot x^{2}) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.6.2

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

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Step 3.1.7.1.6.2.1

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

$3 x^{5} - 9 x^{4} + 3 \cdot 3 (x^{1} x^{2}) + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.6.2.2

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

$3 x^{5} - 9 x^{4} + 3 \cdot 3 x^{1 + 2} + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.6.3

Add $1$ and $2$ .

$3 x^{5} - 9 x^{4} + 3 \cdot 3 x^{3} + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.7

Multiply $3$ by $3$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} + 3 x^{2} \cdot - 1 + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.8

Multiply $- 1$ by $3$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x \cdot x^{3} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.9

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

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Step 3.1.7.1.9.1

Move $x^{3}$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 (x^{3} x) + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.9.2

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

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Step 3.1.7.1.9.2.1

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

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 (x^{3} x^{1}) + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.9.2.2

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

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{3 + 1} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.9.3

Add $3$ and $1$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 x (- 3 x^{2}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.10

Rewrite using the commutative property of multiplication.

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 x \cdot x^{2} + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.11

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

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Step 3.1.7.1.11.1

Move $x^{2}$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 (x^{2} x) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.11.2

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

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Step 3.1.7.1.11.2.1

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

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 (x^{2} x^{1}) + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.11.2.2

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

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 x^{2 + 1} + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.11.3

Add $2$ and $1$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} + 15 \cdot - 3 x^{3} + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.12

Multiply $15$ by $- 3$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 15 x (3 x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.13

Rewrite using the commutative property of multiplication.

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 15 \cdot 3 x \cdot x + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.14

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

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Step 3.1.7.1.14.1

Move $x$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 15 \cdot 3 (x \cdot x) + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.14.2

Multiply $x$ by $x$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 15 \cdot 3 x^{2} + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.15

Multiply $15$ by $3$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} + 15 x \cdot - 1 + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.16

Multiply $- 1$ by $15$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} + 18 (- 3 x^{2}) + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.17

Multiply $- 3$ by $18$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 18 (3 x) + 18 \cdot - 1$

Step 3.1.7.1.18

Multiply $3$ by $18$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 54 x + 18 \cdot - 1$

Step 3.1.7.1.19

Multiply $18$ by $- 1$ .

$3 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 15 x^{4} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 54 x - 18$

Step 3.1.7.2

Simplify by adding terms.

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Step 3.1.7.2.1

Add $- 9 x^{4}$ and $15 x^{4}$ .

$3 x^{5} + 6 x^{4} + 9 x^{3} - 3 x^{2} - 45 x^{3} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 54 x - 18$

Step 3.1.7.2.2

Subtract $45 x^{3}$ from $9 x^{3}$ .

$3 x^{5} + 6 x^{4} - 36 x^{3} - 3 x^{2} + 45 x^{2} - 15 x + 18 x^{3} - 54 x^{2} + 54 x - 18$

Step 3.1.7.2.3

Add $- 36 x^{3}$ and $18 x^{3}$ .

$3 x^{5} + 6 x^{4} - 18 x^{3} - 3 x^{2} + 45 x^{2} - 15 x - 54 x^{2} + 54 x - 18$

Step 3.1.7.2.4

Add $- 3 x^{2}$ and $45 x^{2}$ .

$3 x^{5} + 6 x^{4} - 18 x^{3} + 42 x^{2} - 15 x - 54 x^{2} + 54 x - 18$

Step 3.1.7.2.5

Subtract $54 x^{2}$ from $42 x^{2}$ .

$3 x^{5} + 6 x^{4} - 18 x^{3} - 12 x^{2} - 15 x + 54 x - 18$

Step 3.1.7.2.6

Add $- 15 x$ and $54 x$ .

$3 x^{5} + 6 x^{4} - 18 x^{3} - 12 x^{2} + 39 x - 18$

Step 3.2

The leading term in a polynomial is the term with the highest degree.

$3 x^{5}$

Step 3.3

The leading coefficient in a polynomial is the coefficient of the leading term.

$3$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Falls to the left and rises to the right

Step 7