Algebra Examples

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

Step 1

Write $\frac{1}{4} (3 x - 2) (x^{2} - 5) {(x + 6)}^{2}$ as a function.

$f (x) = \frac{1}{4} (3 x - 2) (x^{2} - 5) {(x + 6)}^{2}$

Step 2

Identify the degree of the function.

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Step 2.1

Simplify and reorder the polynomial.

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Step 2.1.1

Expand $(3 x - 2) (x^{2} - 5)$ using the FOIL Method.

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Step 2.1.1.1

Apply the distributive property.

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

Step 2.1.1.2

Apply the distributive property.

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

Step 2.1.1.3

Apply the distributive property.

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

Step 2.1.2

Simplify terms.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

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

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Step 2.1.2.1.1.1

Move $x^{2}$ .

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

Step 2.1.2.1.1.2

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

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Step 2.1.2.1.1.2.1

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

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

Step 2.1.2.1.1.2.2

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

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

Step 2.1.2.1.1.3

Add $2$ and $1$ .

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

Step 2.1.2.1.2

Multiply $- 5$ by $3$ .

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

Step 2.1.2.1.3

Multiply $- 2$ by $- 5$ .

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) {(x + 6)}^{2})$

Step 2.1.2.2

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

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) ((x + 6) (x + 6)))$

Step 2.1.3

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

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Step 2.1.3.1

Apply the distributive property.

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x (x + 6) + 6 (x + 6)))$

Step 2.1.3.2

Apply the distributive property.

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x \cdot x + x \cdot 6 + 6 (x + 6)))$

Step 2.1.3.3

Apply the distributive property.

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6))$

Step 2.1.4

Simplify and combine like terms.

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Step 2.1.4.1

Simplify each term.

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Step 2.1.4.1.1

Multiply $x$ by $x$ .

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + x \cdot 6 + 6 x + 6 \cdot 6))$

Step 2.1.4.1.2

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

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + 6 \cdot x + 6 x + 6 \cdot 6))$

Step 2.1.4.1.3

Multiply $6$ by $6$ .

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + 6 x + 6 x + 36))$

Step 2.1.4.2

Add $6 x$ and $6 x$ .

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + 12 x + 36))$

Step 2.1.5

Expand $(3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + 12 x + 36)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{1}{4} \cdot (3 x^{3} x^{2} + 3 x^{3} (12 x) + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6

Simplify terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

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

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Step 2.1.6.1.1.1

Move $x^{2}$ .

$\frac{1}{4} \cdot (3 (x^{2} x^{3}) + 3 x^{3} (12 x) + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.1.2

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

$\frac{1}{4} \cdot (3 x^{2 + 3} + 3 x^{3} (12 x) + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.1.3

Add $2$ and $3$ .

$\frac{1}{4} \cdot (3 x^{5} + 3 x^{3} (12 x) + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.2

Rewrite using the commutative property of multiplication.

$\frac{1}{4} \cdot (3 x^{5} + 3 \cdot 12 x^{3} x + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.3

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

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Step 2.1.6.1.3.1

Move $x$ .

$\frac{1}{4} \cdot (3 x^{5} + 3 \cdot 12 (x \cdot x^{3}) + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.3.2

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

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Step 2.1.6.1.3.2.1

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

$\frac{1}{4} \cdot (3 x^{5} + 3 \cdot 12 (x^{1} x^{3}) + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.3.2.2

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

$\frac{1}{4} \cdot (3 x^{5} + 3 \cdot 12 x^{1 + 3} + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.3.3

Add $1$ and $3$ .

$\frac{1}{4} \cdot (3 x^{5} + 3 \cdot 12 x^{4} + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.4

Multiply $3$ by $12$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.5

Multiply $36$ by $3$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.6

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

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Step 2.1.6.1.6.1

Move $x^{2}$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 (x^{2} x) - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.6.2

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

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Step 2.1.6.1.6.2.1

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 (x^{2} x^{1}) - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.6.2.2

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{2 + 1} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.6.3

Add $2$ and $1$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.7

Rewrite using the commutative property of multiplication.

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 15 \cdot 12 x \cdot x - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.8

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

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Step 2.1.6.1.8.1

Move $x$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 15 \cdot 12 (x \cdot x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.8.2

Multiply $x$ by $x$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 15 \cdot 12 x^{2} - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.9

Multiply $- 15$ by $12$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.10

Multiply $36$ by $- 15$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.11

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

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Step 2.1.6.1.11.1

Move $x^{2}$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 (x^{2} x^{2}) - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.11.2

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{2 + 2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.11.3

Add $2$ and $2$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.12

Rewrite using the commutative property of multiplication.

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 x^{2} x - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.13

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

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Step 2.1.6.1.13.1

Move $x$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 (x \cdot x^{2}) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.13.2

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

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Step 2.1.6.1.13.2.1

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 (x^{1} x^{2}) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.13.2.2

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 x^{1 + 2} - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.13.3

Add $1$ and $2$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 x^{3} - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.14

Multiply $- 2$ by $12$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 24 x^{3} - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.15

Multiply $36$ by $- 2$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 24 x^{3} - 72 x^{2} + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 2.1.6.1.16

Multiply $12$ by $10$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 24 x^{3} - 72 x^{2} + 10 x^{2} + 120 x + 10 \cdot 36)$

Step 2.1.6.1.17

Multiply $10$ by $36$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 24 x^{3} - 72 x^{2} + 10 x^{2} + 120 x + 360)$

Step 2.1.6.2

Simplify terms.

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Step 2.1.6.2.1

Subtract $2 x^{4}$ from $36 x^{4}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 24 x^{3} - 72 x^{2} + 10 x^{2} + 120 x + 360)$

Step 2.1.6.2.2

Subtract $15 x^{3}$ from $108 x^{3}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 93 x^{3} - 180 x^{2} - 540 x - 24 x^{3} - 72 x^{2} + 10 x^{2} + 120 x + 360)$

Step 2.1.6.2.3

Subtract $24 x^{3}$ from $93 x^{3}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 69 x^{3} - 180 x^{2} - 540 x - 72 x^{2} + 10 x^{2} + 120 x + 360)$

Step 2.1.6.2.4

Subtract $72 x^{2}$ from $- 180 x^{2}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 69 x^{3} - 252 x^{2} - 540 x + 10 x^{2} + 120 x + 360)$

Step 2.1.6.2.5

Add $- 252 x^{2}$ and $10 x^{2}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 69 x^{3} - 242 x^{2} - 540 x + 120 x + 360)$

Step 2.1.6.2.6

Add $- 540 x$ and $120 x$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 69 x^{3} - 242 x^{2} - 420 x + 360)$

Step 2.1.6.2.7

Apply the distributive property.

$\frac{1}{4} (3 x^{5}) + \frac{1}{4} (34 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7

Simplify.

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Step 2.1.7.1

Multiply $\frac{1}{4} (3 x^{5})$ .

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Step 2.1.7.1.1

Combine $3$ and $\frac{1}{4}$ .

$\frac{3}{4} x^{5} + \frac{1}{4} (34 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.1.2

Combine $\frac{3}{4}$ and $x^{5}$ .

$\frac{3 x^{5}}{4} + \frac{1}{4} (34 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.2

Cancel the common factor of $2$ .

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Step 2.1.7.2.1

Factor $2$ out of $4$ .

$\frac{3 x^{5}}{4} + \frac{1}{2 (2)} (34 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.2.2

Factor $2$ out of $34 x^{4}$ .

$\frac{3 x^{5}}{4} + \frac{1}{2 (2)} (2 (17 x^{4})) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.2.3

Cancel the common factor.

$\frac{3 x^{5}}{4} + \frac{1}{2 \cdot 2} (2 (17 x^{4})) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.2.4

Rewrite the expression.

$\frac{3 x^{5}}{4} + \frac{1}{2} (17 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.3

Combine $17$ and $\frac{1}{2}$ .

$\frac{3 x^{5}}{4} + \frac{17}{2} x^{4} + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.4

Combine $\frac{17}{2}$ and $x^{4}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.5

Multiply $\frac{1}{4} (69 x^{3})$ .

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Step 2.1.7.5.1

Combine $69$ and $\frac{1}{4}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69}{4} x^{3} + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.5.2

Combine $\frac{69}{4}$ and $x^{3}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.6

Cancel the common factor of $2$ .

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Step 2.1.7.6.1

Factor $2$ out of $4$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{2 (2)} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.6.2

Factor $2$ out of $- 242 x^{2}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{2 (2)} (2 (- 121 x^{2})) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.6.3

Cancel the common factor.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{2 \cdot 2} (2 (- 121 x^{2})) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.6.4

Rewrite the expression.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{2} (- 121 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.7

Combine $- 121$ and $\frac{1}{2}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121}{2} x^{2} + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.8

Combine $\frac{- 121}{2}$ and $x^{2}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 2.1.7.9

Cancel the common factor of $4$ .

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Step 2.1.7.9.1

Factor $4$ out of $- 420 x$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} + \frac{1}{4} (4 (- 105 x)) + \frac{1}{4} \cdot 360$

Step 2.1.7.9.2

Cancel the common factor.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} + \frac{1}{4} (4 (- 105 x)) + \frac{1}{4} \cdot 360$

Step 2.1.7.9.3

Rewrite the expression.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} - 105 x + \frac{1}{4} \cdot 360$

Step 2.1.7.10

Cancel the common factor of $4$ .

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Step 2.1.7.10.1

Factor $4$ out of $360$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} - 105 x + \frac{1}{4} \cdot (4 (90))$

Step 2.1.7.10.2

Cancel the common factor.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} - 105 x + \frac{1}{4} \cdot (4 \cdot 90)$

Step 2.1.7.10.3

Rewrite the expression.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} - 105 x + 90$

Step 2.1.8

Move the negative in front of the fraction.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} - \frac{121 x^{2}}{2} - 105 x + 90$

Step 2.2

The largest exponent is the degree of the polynomial.

$5$

Step 3

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

Odd

Step 4

Identify the leading coefficient.

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Step 4.1

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

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Step 4.1.1

Expand $(3 x - 2) (x^{2} - 5)$ using the FOIL Method.

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Step 4.1.1.1

Apply the distributive property.

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

Step 4.1.1.2

Apply the distributive property.

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

Step 4.1.1.3

Apply the distributive property.

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

Step 4.1.2

Simplify terms.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

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Step 4.1.2.1.1.1

Move $x^{2}$ .

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

Step 4.1.2.1.1.2

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

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Step 4.1.2.1.1.2.1

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

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

Step 4.1.2.1.1.2.2

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

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

Step 4.1.2.1.1.3

Add $2$ and $1$ .

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

Step 4.1.2.1.2

Multiply $- 5$ by $3$ .

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

Step 4.1.2.1.3

Multiply $- 2$ by $- 5$ .

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) {(x + 6)}^{2})$

Step 4.1.2.2

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

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) ((x + 6) (x + 6)))$

Step 4.1.3

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

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Step 4.1.3.1

Apply the distributive property.

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x (x + 6) + 6 (x + 6)))$

Step 4.1.3.2

Apply the distributive property.

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x \cdot x + x \cdot 6 + 6 (x + 6)))$

Step 4.1.3.3

Apply the distributive property.

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6))$

Step 4.1.4

Simplify and combine like terms.

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Step 4.1.4.1

Simplify each term.

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Step 4.1.4.1.1

Multiply $x$ by $x$ .

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + x \cdot 6 + 6 x + 6 \cdot 6))$

Step 4.1.4.1.2

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

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + 6 \cdot x + 6 x + 6 \cdot 6))$

Step 4.1.4.1.3

Multiply $6$ by $6$ .

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + 6 x + 6 x + 36))$

Step 4.1.4.2

Add $6 x$ and $6 x$ .

$\frac{1}{4} \cdot ((3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + 12 x + 36))$

Step 4.1.5

Expand $(3 x^{3} - 15 x - 2 x^{2} + 10) (x^{2} + 12 x + 36)$ by multiplying each term in the first expression by each term in the second expression.

Step 4.1.6

Simplify terms.

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Step 4.1.6.1

Simplify each term.

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Step 4.1.6.1.1

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

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Step 4.1.6.1.1.1

Move $x^{2}$ .

Step 4.1.6.1.1.2

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

Step 4.1.6.1.1.3

Add $2$ and $3$ .

Step 4.1.6.1.2

Rewrite using the commutative property of multiplication.

Step 4.1.6.1.3

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

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Step 4.1.6.1.3.1

Move $x$ .

Step 4.1.6.1.3.2

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

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Step 4.1.6.1.3.2.1

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

Step 4.1.6.1.3.2.2

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

Step 4.1.6.1.3.3

Add $1$ and $3$ .

Step 4.1.6.1.4

Multiply $3$ by $12$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 3 x^{3} \cdot 36 - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.5

Multiply $36$ by $3$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x \cdot x^{2} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.6

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

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Step 4.1.6.1.6.1

Move $x^{2}$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 (x^{2} x) - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.6.2

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

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Step 4.1.6.1.6.2.1

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 (x^{2} x^{1}) - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.6.2.2

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{2 + 1} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.6.3

Add $2$ and $1$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 15 x (12 x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.7

Rewrite using the commutative property of multiplication.

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 15 \cdot 12 x \cdot x - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.8

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

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Step 4.1.6.1.8.1

Move $x$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 15 \cdot 12 (x \cdot x) - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.8.2

Multiply $x$ by $x$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 15 \cdot 12 x^{2} - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.9

Multiply $- 15$ by $12$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 15 x \cdot 36 - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.10

Multiply $36$ by $- 15$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{2} x^{2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.11

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

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Step 4.1.6.1.11.1

Move $x^{2}$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 (x^{2} x^{2}) - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.11.2

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{2 + 2} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.11.3

Add $2$ and $2$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 x^{2} (12 x) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.12

Rewrite using the commutative property of multiplication.

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 x^{2} x - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.13

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

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Step 4.1.6.1.13.1

Move $x$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 (x \cdot x^{2}) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.13.2

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

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Step 4.1.6.1.13.2.1

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 (x^{1} x^{2}) - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.13.2.2

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

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 x^{1 + 2} - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.13.3

Add $1$ and $2$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 2 \cdot 12 x^{3} - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.14

Multiply $- 2$ by $12$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 24 x^{3} - 2 x^{2} \cdot 36 + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.15

Multiply $36$ by $- 2$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 24 x^{3} - 72 x^{2} + 10 x^{2} + 10 (12 x) + 10 \cdot 36)$

Step 4.1.6.1.16

Multiply $12$ by $10$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 24 x^{3} - 72 x^{2} + 10 x^{2} + 120 x + 10 \cdot 36)$

Step 4.1.6.1.17

Multiply $10$ by $36$ .

$\frac{1}{4} \cdot (3 x^{5} + 36 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 2 x^{4} - 24 x^{3} - 72 x^{2} + 10 x^{2} + 120 x + 360)$

Step 4.1.6.2

Simplify terms.

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Step 4.1.6.2.1

Subtract $2 x^{4}$ from $36 x^{4}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 108 x^{3} - 15 x^{3} - 180 x^{2} - 540 x - 24 x^{3} - 72 x^{2} + 10 x^{2} + 120 x + 360)$

Step 4.1.6.2.2

Subtract $15 x^{3}$ from $108 x^{3}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 93 x^{3} - 180 x^{2} - 540 x - 24 x^{3} - 72 x^{2} + 10 x^{2} + 120 x + 360)$

Step 4.1.6.2.3

Subtract $24 x^{3}$ from $93 x^{3}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 69 x^{3} - 180 x^{2} - 540 x - 72 x^{2} + 10 x^{2} + 120 x + 360)$

Step 4.1.6.2.4

Subtract $72 x^{2}$ from $- 180 x^{2}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 69 x^{3} - 252 x^{2} - 540 x + 10 x^{2} + 120 x + 360)$

Step 4.1.6.2.5

Add $- 252 x^{2}$ and $10 x^{2}$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 69 x^{3} - 242 x^{2} - 540 x + 120 x + 360)$

Step 4.1.6.2.6

Add $- 540 x$ and $120 x$ .

$\frac{1}{4} \cdot (3 x^{5} + 34 x^{4} + 69 x^{3} - 242 x^{2} - 420 x + 360)$

Step 4.1.6.2.7

Apply the distributive property.

$\frac{1}{4} (3 x^{5}) + \frac{1}{4} (34 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7

Simplify.

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Step 4.1.7.1

Multiply $\frac{1}{4} (3 x^{5})$ .

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Step 4.1.7.1.1

Combine $3$ and $\frac{1}{4}$ .

$\frac{3}{4} x^{5} + \frac{1}{4} (34 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.1.2

Combine $\frac{3}{4}$ and $x^{5}$ .

$\frac{3 x^{5}}{4} + \frac{1}{4} (34 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.2

Cancel the common factor of $2$ .

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Step 4.1.7.2.1

Factor $2$ out of $4$ .

$\frac{3 x^{5}}{4} + \frac{1}{2 (2)} (34 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.2.2

Factor $2$ out of $34 x^{4}$ .

$\frac{3 x^{5}}{4} + \frac{1}{2 (2)} (2 (17 x^{4})) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.2.3

Cancel the common factor.

$\frac{3 x^{5}}{4} + \frac{1}{2 \cdot 2} (2 (17 x^{4})) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.2.4

Rewrite the expression.

$\frac{3 x^{5}}{4} + \frac{1}{2} (17 x^{4}) + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.3

Combine $17$ and $\frac{1}{2}$ .

$\frac{3 x^{5}}{4} + \frac{17}{2} x^{4} + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.4

Combine $\frac{17}{2}$ and $x^{4}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{1}{4} (69 x^{3}) + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.5

Multiply $\frac{1}{4} (69 x^{3})$ .

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Step 4.1.7.5.1

Combine $69$ and $\frac{1}{4}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69}{4} x^{3} + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.5.2

Combine $\frac{69}{4}$ and $x^{3}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{4} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.6

Cancel the common factor of $2$ .

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Step 4.1.7.6.1

Factor $2$ out of $4$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{2 (2)} (- 242 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.6.2

Factor $2$ out of $- 242 x^{2}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{2 (2)} (2 (- 121 x^{2})) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.6.3

Cancel the common factor.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{2 \cdot 2} (2 (- 121 x^{2})) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.6.4

Rewrite the expression.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{1}{2} (- 121 x^{2}) + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.7

Combine $- 121$ and $\frac{1}{2}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121}{2} x^{2} + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.8

Combine $\frac{- 121}{2}$ and $x^{2}$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} + \frac{1}{4} (- 420 x) + \frac{1}{4} \cdot 360$

Step 4.1.7.9

Cancel the common factor of $4$ .

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Step 4.1.7.9.1

Factor $4$ out of $- 420 x$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} + \frac{1}{4} (4 (- 105 x)) + \frac{1}{4} \cdot 360$

Step 4.1.7.9.2

Cancel the common factor.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} + \frac{1}{4} (4 (- 105 x)) + \frac{1}{4} \cdot 360$

Step 4.1.7.9.3

Rewrite the expression.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} - 105 x + \frac{1}{4} \cdot 360$

Step 4.1.7.10

Cancel the common factor of $4$ .

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Step 4.1.7.10.1

Factor $4$ out of $360$ .

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} - 105 x + \frac{1}{4} \cdot (4 (90))$

Step 4.1.7.10.2

Cancel the common factor.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} - 105 x + \frac{1}{4} \cdot (4 \cdot 90)$

Step 4.1.7.10.3

Rewrite the expression.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} + \frac{- 121 x^{2}}{2} - 105 x + 90$

Step 4.1.8

Move the negative in front of the fraction.

$\frac{3 x^{5}}{4} + \frac{17 x^{4}}{2} + \frac{69 x^{3}}{4} - \frac{121 x^{2}}{2} - 105 x + 90$

Step 4.2

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

$\frac{3 x^{5}}{4}$

Step 4.3

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

$\frac{3}{4}$

Step 5

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

Positive

Step 6

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 7

Determine the behavior.

Falls to the left and rises to the right

Step 8