Algebra Examples

$f (x) = - \frac{3}{4} x (3 x - 2) (2 x + 7) (x + 5) (x - 1)$

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 terms.

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Step 1.1.1.1

Apply the distributive property.

$- \frac{3}{4} \cdot ((x (3 x) + x \cdot - 2) (2 x + 7) (x + 5) (x - 1))$

Step 1.1.1.2

Reorder.

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Step 1.1.1.2.1

Rewrite using the commutative property of multiplication.

$- \frac{3}{4} \cdot ((3 x \cdot x + x \cdot - 2) (2 x + 7) (x + 5) (x - 1))$

Step 1.1.1.2.2

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

$- \frac{3}{4} \cdot ((3 x \cdot x - 2 \cdot x) (2 x + 7) (x + 5) (x - 1))$

Step 1.1.1.3

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

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Step 1.1.1.3.1

Move $x$ .

$- \frac{3}{4} \cdot ((3 (x \cdot x) - 2 \cdot x) (2 x + 7) (x + 5) (x - 1))$

Step 1.1.1.3.2

Multiply $x$ by $x$ .

$- \frac{3}{4} \cdot ((3 x^{2} - 2 \cdot x) (2 x + 7) (x + 5) (x - 1))$

$- \frac{3}{4} \cdot ((3 x^{2} - 2 x) (2 x + 7) (x + 5) (x - 1))$

Step 1.1.2

Expand $(3 x^{2} - 2 x) (2 x + 7)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$- \frac{3}{4} \cdot ((3 x^{2} (2 x + 7) - 2 x (2 x + 7)) (x + 5) (x - 1))$

Step 1.1.2.2

Apply the distributive property.

$- \frac{3}{4} \cdot ((3 x^{2} (2 x) + 3 x^{2} \cdot 7 - 2 x (2 x + 7)) (x + 5) (x - 1))$

Step 1.1.2.3

Apply the distributive property.

$- \frac{3}{4} \cdot ((3 x^{2} (2 x) + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

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

Rewrite using the commutative property of multiplication.

$- \frac{3}{4} \cdot ((3 \cdot 2 x^{2} x + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.2

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

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Step 1.1.3.1.2.1

Move $x$ .

$- \frac{3}{4} \cdot ((3 \cdot 2 (x \cdot x^{2}) + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.2.2

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

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Step 1.1.3.1.2.2.1

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

$- \frac{3}{4} \cdot ((3 \cdot 2 (x^{1} x^{2}) + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.2.2.2

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

$- \frac{3}{4} \cdot ((3 \cdot 2 x^{1 + 2} + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.2.3

Add $1$ and $2$ .

$- \frac{3}{4} \cdot ((3 \cdot 2 x^{3} + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.3

Multiply $3$ by $2$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.4

Multiply $7$ by $3$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.5

Rewrite using the commutative property of multiplication.

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 2 \cdot 2 x \cdot x - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.6

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

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Step 1.1.3.1.6.1

Move $x$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 2 \cdot 2 (x \cdot x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.6.2

Multiply $x$ by $x$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 2 \cdot 2 x^{2} - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.7

Multiply $- 2$ by $2$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 4 x^{2} - 2 x \cdot 7) (x + 5) (x - 1))$

Step 1.1.3.1.8

Multiply $7$ by $- 2$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 4 x^{2} - 14 x) (x + 5) (x - 1))$

Step 1.1.3.2

Subtract $4 x^{2}$ from $21 x^{2}$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 17 x^{2} - 14 x) (x + 5) (x - 1))$

Step 1.1.4

Expand $(6 x^{3} + 17 x^{2} - 14 x) (x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{3}{4} \cdot ((6 x^{3} x + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5

Simplify terms.

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Step 1.1.5.1

Simplify each term.

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Step 1.1.5.1.1

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

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Step 1.1.5.1.1.1

Move $x$ .

$- \frac{3}{4} \cdot ((6 (x \cdot x^{3}) + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.1.2

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

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Step 1.1.5.1.1.2.1

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

$- \frac{3}{4} \cdot ((6 (x^{1} x^{3}) + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.1.2.2

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

$- \frac{3}{4} \cdot ((6 x^{1 + 3} + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.1.3

Add $1$ and $3$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.2

Multiply $5$ by $6$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.3

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

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Step 1.1.5.1.3.1

Move $x$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 (x \cdot x^{2}) + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.3.2

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

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Step 1.1.5.1.3.2.1

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

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 (x^{1} x^{2}) + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.3.2.2

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

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{1 + 2} + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.3.3

Add $1$ and $2$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.4

Multiply $5$ by $17$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 85 x^{2} - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.5

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

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Step 1.1.5.1.5.1

Move $x$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 85 x^{2} - 14 (x \cdot x) - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.5.2

Multiply $x$ by $x$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 85 x^{2} - 14 x^{2} - 14 x \cdot 5) (x - 1))$

Step 1.1.5.1.6

Multiply $5$ by $- 14$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 85 x^{2} - 14 x^{2} - 70 x) (x - 1))$

Step 1.1.5.2

Simplify by adding terms.

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Step 1.1.5.2.1

Add $30 x^{3}$ and $17 x^{3}$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 47 x^{3} + 85 x^{2} - 14 x^{2} - 70 x) (x - 1))$

Step 1.1.5.2.2

Subtract $14 x^{2}$ from $85 x^{2}$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 47 x^{3} + 71 x^{2} - 70 x) (x - 1))$

Step 1.1.6

Expand $(6 x^{4} + 47 x^{3} + 71 x^{2} - 70 x) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{3}{4} \cdot (6 x^{4} x + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

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

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Step 1.1.7.1.1.1

Move $x$ .

$- \frac{3}{4} \cdot (6 (x \cdot x^{4}) + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.1.2

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

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Step 1.1.7.1.1.2.1

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

$- \frac{3}{4} \cdot (6 (x^{1} x^{4}) + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.1.2.2

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

$- \frac{3}{4} \cdot (6 x^{1 + 4} + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.1.3

Add $1$ and $4$ .

$- \frac{3}{4} \cdot (6 x^{5} + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.2

Multiply $- 1$ by $6$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.3

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

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Step 1.1.7.1.3.1

Move $x$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 (x \cdot x^{3}) + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.3.2

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

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Step 1.1.7.1.3.2.1

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

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 (x^{1} x^{3}) + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.3.2.2

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

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{1 + 3} + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.3.3

Add $1$ and $3$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.4

Multiply $- 1$ by $47$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.5

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

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Step 1.1.7.1.5.1

Move $x$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 (x \cdot x^{2}) + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.5.2

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

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Step 1.1.7.1.5.2.1

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

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 (x^{1} x^{2}) + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.5.2.2

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

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{1 + 2} + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.5.3

Add $1$ and $2$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.6

Multiply $- 1$ by $71$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 x \cdot x - 70 x \cdot - 1)$

Step 1.1.7.1.7

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

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Step 1.1.7.1.7.1

Move $x$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 (x \cdot x) - 70 x \cdot - 1)$

Step 1.1.7.1.7.2

Multiply $x$ by $x$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 x^{2} - 70 x \cdot - 1)$

Step 1.1.7.1.8

Multiply $- 1$ by $- 70$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 x^{2} + 70 x)$

Step 1.1.7.2

Simplify terms.

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Step 1.1.7.2.1

Add $- 6 x^{4}$ and $47 x^{4}$ .

$- \frac{3}{4} \cdot (6 x^{5} + 41 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 x^{2} + 70 x)$

Step 1.1.7.2.2

Add $- 47 x^{3}$ and $71 x^{3}$ .

$- \frac{3}{4} \cdot (6 x^{5} + 41 x^{4} + 24 x^{3} - 71 x^{2} - 70 x^{2} + 70 x)$

Step 1.1.7.2.3

Subtract $70 x^{2}$ from $- 71 x^{2}$ .

$- \frac{3}{4} \cdot (6 x^{5} + 41 x^{4} + 24 x^{3} - 141 x^{2} + 70 x)$

Step 1.1.7.2.4

Apply the distributive property.

$- \frac{3}{4} (6 x^{5}) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8

Simplify.

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Step 1.1.8.1

Cancel the common factor of $2$ .

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Step 1.1.8.1.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 3}{4} (6 x^{5}) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.1.2

Factor $2$ out of $4$ .

$\frac{- 3}{2 (2)} (6 x^{5}) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.1.3

Factor $2$ out of $6 x^{5}$ .

$\frac{- 3}{2 (2)} (2 (3 x^{5})) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.1.4

Cancel the common factor.

$\frac{- 3}{2 \cdot 2} (2 (3 x^{5})) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.1.5

Rewrite the expression.

$\frac{- 3}{2} (3 x^{5}) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.2

Combine $3$ and $\frac{- 3}{2}$ .

$\frac{3 \cdot - 3}{2} x^{5} - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.3

Multiply $3$ by $- 3$ .

$\frac{- 9}{2} x^{5} - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.4

Combine $\frac{- 9}{2}$ and $x^{5}$ .

$\frac{- 9 x^{5}}{2} - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.5

Multiply $- \frac{3}{4} (41 x^{4})$ .

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Step 1.1.8.5.1

Multiply $41$ by $- 1$ .

$\frac{- 9 x^{5}}{2} - 41 (\frac{3}{4}) x^{4} - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.5.2

Combine $- 41$ and $\frac{3}{4}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 41 \cdot 3}{4} x^{4} - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.5.3

Multiply $- 41$ by $3$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123}{4} x^{4} - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.5.4

Combine $\frac{- 123}{4}$ and $x^{4}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.6

Cancel the common factor of $4$ .

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Step 1.1.8.6.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} + \frac{- 3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.6.2

Factor $4$ out of $24 x^{3}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} + \frac{- 3}{4} (4 (6 x^{3})) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.6.3

Cancel the common factor.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} + \frac{- 3}{4} (4 (6 x^{3})) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.6.4

Rewrite the expression.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 3 (6 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.7

Multiply $6$ by $- 3$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 1.1.8.8

Multiply $- \frac{3}{4} (- 141 x^{2})$ .

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Step 1.1.8.8.1

Multiply $- 141$ by $- 1$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + 141 (\frac{3}{4}) x^{2} - \frac{3}{4} (70 x)$

Step 1.1.8.8.2

Combine $141$ and $\frac{3}{4}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{141 \cdot 3}{4} x^{2} - \frac{3}{4} (70 x)$

Step 1.1.8.8.3

Multiply $141$ by $3$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423}{4} x^{2} - \frac{3}{4} (70 x)$

Step 1.1.8.8.4

Combine $\frac{423}{4}$ and $x^{2}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} - \frac{3}{4} (70 x)$

Step 1.1.8.9

Cancel the common factor of $2$ .

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Step 1.1.8.9.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{4} (70 x)$

Step 1.1.8.9.2

Factor $2$ out of $4$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{2 (2)} (70 x)$

Step 1.1.8.9.3

Factor $2$ out of $70 x$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{2 (2)} (2 (35 x))$

Step 1.1.8.9.4

Cancel the common factor.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{2 \cdot 2} (2 (35 x))$

Step 1.1.8.9.5

Rewrite the expression.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{2} (35 x)$

Step 1.1.8.10

Combine $35$ and $\frac{- 3}{2}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{35 \cdot - 3}{2} x$

Step 1.1.8.11

Multiply $35$ by $- 3$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 105}{2} x$

Step 1.1.8.12

Combine $\frac{- 105}{2}$ and $x$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 105 x}{2}$

Step 1.1.9

Simplify each term.

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Step 1.1.9.1

Move the negative in front of the fraction.

$- \frac{9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 105 x}{2}$

Step 1.1.9.2

Move the negative in front of the fraction.

$- \frac{9 x^{5}}{2} - \frac{123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 105 x}{2}$

Step 1.1.9.3

Move the negative in front of the fraction.

$- \frac{9 x^{5}}{2} - \frac{123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} - \frac{105 x}{2}$

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 terms.

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Step 3.1.1.1

Apply the distributive property.

$- \frac{3}{4} \cdot ((x (3 x) + x \cdot - 2) (2 x + 7) (x + 5) (x - 1))$

Step 3.1.1.2

Reorder.

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Step 3.1.1.2.1

Rewrite using the commutative property of multiplication.

$- \frac{3}{4} \cdot ((3 x \cdot x + x \cdot - 2) (2 x + 7) (x + 5) (x - 1))$

Step 3.1.1.2.2

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

$- \frac{3}{4} \cdot ((3 x \cdot x - 2 \cdot x) (2 x + 7) (x + 5) (x - 1))$

Step 3.1.1.3

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

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Step 3.1.1.3.1

Move $x$ .

$- \frac{3}{4} \cdot ((3 (x \cdot x) - 2 \cdot x) (2 x + 7) (x + 5) (x - 1))$

Step 3.1.1.3.2

Multiply $x$ by $x$ .

$- \frac{3}{4} \cdot ((3 x^{2} - 2 \cdot x) (2 x + 7) (x + 5) (x - 1))$

$- \frac{3}{4} \cdot ((3 x^{2} - 2 x) (2 x + 7) (x + 5) (x - 1))$

Step 3.1.2

Expand $(3 x^{2} - 2 x) (2 x + 7)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$- \frac{3}{4} \cdot ((3 x^{2} (2 x + 7) - 2 x (2 x + 7)) (x + 5) (x - 1))$

Step 3.1.2.2

Apply the distributive property.

$- \frac{3}{4} \cdot ((3 x^{2} (2 x) + 3 x^{2} \cdot 7 - 2 x (2 x + 7)) (x + 5) (x - 1))$

Step 3.1.2.3

Apply the distributive property.

$- \frac{3}{4} \cdot ((3 x^{2} (2 x) + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

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

Rewrite using the commutative property of multiplication.

$- \frac{3}{4} \cdot ((3 \cdot 2 x^{2} x + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.2

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

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Step 3.1.3.1.2.1

Move $x$ .

$- \frac{3}{4} \cdot ((3 \cdot 2 (x \cdot x^{2}) + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.2.2

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

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Step 3.1.3.1.2.2.1

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

$- \frac{3}{4} \cdot ((3 \cdot 2 (x^{1} x^{2}) + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.2.2.2

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

$- \frac{3}{4} \cdot ((3 \cdot 2 x^{1 + 2} + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.2.3

Add $1$ and $2$ .

$- \frac{3}{4} \cdot ((3 \cdot 2 x^{3} + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.3

Multiply $3$ by $2$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 3 x^{2} \cdot 7 - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.4

Multiply $7$ by $3$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 2 x (2 x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.5

Rewrite using the commutative property of multiplication.

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 2 \cdot 2 x \cdot x - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.6

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

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Step 3.1.3.1.6.1

Move $x$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 2 \cdot 2 (x \cdot x) - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.6.2

Multiply $x$ by $x$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 2 \cdot 2 x^{2} - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.7

Multiply $- 2$ by $2$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 4 x^{2} - 2 x \cdot 7) (x + 5) (x - 1))$

Step 3.1.3.1.8

Multiply $7$ by $- 2$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 21 x^{2} - 4 x^{2} - 14 x) (x + 5) (x - 1))$

Step 3.1.3.2

Subtract $4 x^{2}$ from $21 x^{2}$ .

$- \frac{3}{4} \cdot ((6 x^{3} + 17 x^{2} - 14 x) (x + 5) (x - 1))$

Step 3.1.4

Expand $(6 x^{3} + 17 x^{2} - 14 x) (x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{3}{4} \cdot ((6 x^{3} x + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5

Simplify terms.

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Step 3.1.5.1

Simplify each term.

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Step 3.1.5.1.1

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

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Step 3.1.5.1.1.1

Move $x$ .

$- \frac{3}{4} \cdot ((6 (x \cdot x^{3}) + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.1.2

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

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Step 3.1.5.1.1.2.1

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

$- \frac{3}{4} \cdot ((6 (x^{1} x^{3}) + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.1.2.2

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

$- \frac{3}{4} \cdot ((6 x^{1 + 3} + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.1.3

Add $1$ and $3$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 6 x^{3} \cdot 5 + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.2

Multiply $5$ by $6$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{2} x + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.3

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

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Step 3.1.5.1.3.1

Move $x$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 (x \cdot x^{2}) + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.3.2

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

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Step 3.1.5.1.3.2.1

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

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 (x^{1} x^{2}) + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.3.2.2

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

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{1 + 2} + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.3.3

Add $1$ and $2$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 17 x^{2} \cdot 5 - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.4

Multiply $5$ by $17$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 85 x^{2} - 14 x \cdot x - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.5

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

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Step 3.1.5.1.5.1

Move $x$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 85 x^{2} - 14 (x \cdot x) - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.5.2

Multiply $x$ by $x$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 85 x^{2} - 14 x^{2} - 14 x \cdot 5) (x - 1))$

Step 3.1.5.1.6

Multiply $5$ by $- 14$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 30 x^{3} + 17 x^{3} + 85 x^{2} - 14 x^{2} - 70 x) (x - 1))$

Step 3.1.5.2

Simplify by adding terms.

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Step 3.1.5.2.1

Add $30 x^{3}$ and $17 x^{3}$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 47 x^{3} + 85 x^{2} - 14 x^{2} - 70 x) (x - 1))$

Step 3.1.5.2.2

Subtract $14 x^{2}$ from $85 x^{2}$ .

$- \frac{3}{4} \cdot ((6 x^{4} + 47 x^{3} + 71 x^{2} - 70 x) (x - 1))$

Step 3.1.6

Expand $(6 x^{4} + 47 x^{3} + 71 x^{2} - 70 x) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{3}{4} \cdot (6 x^{4} x + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

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

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Step 3.1.7.1.1.1

Move $x$ .

$- \frac{3}{4} \cdot (6 (x \cdot x^{4}) + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.1.2

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

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Step 3.1.7.1.1.2.1

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

$- \frac{3}{4} \cdot (6 (x^{1} x^{4}) + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.1.2.2

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

$- \frac{3}{4} \cdot (6 x^{1 + 4} + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.1.3

Add $1$ and $4$ .

$- \frac{3}{4} \cdot (6 x^{5} + 6 x^{4} \cdot - 1 + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.2

Multiply $- 1$ by $6$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{3} x + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.3

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

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Step 3.1.7.1.3.1

Move $x$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 (x \cdot x^{3}) + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.3.2

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

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Step 3.1.7.1.3.2.1

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

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 (x^{1} x^{3}) + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.3.2.2

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

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{1 + 3} + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.3.3

Add $1$ and $3$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} + 47 x^{3} \cdot - 1 + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.4

Multiply $- 1$ by $47$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{2} x + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.5

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

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Step 3.1.7.1.5.1

Move $x$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 (x \cdot x^{2}) + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.5.2

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

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Step 3.1.7.1.5.2.1

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

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 (x^{1} x^{2}) + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.5.2.2

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

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{1 + 2} + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.5.3

Add $1$ and $2$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} + 71 x^{2} \cdot - 1 - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.6

Multiply $- 1$ by $71$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 x \cdot x - 70 x \cdot - 1)$

Step 3.1.7.1.7

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

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Step 3.1.7.1.7.1

Move $x$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 (x \cdot x) - 70 x \cdot - 1)$

Step 3.1.7.1.7.2

Multiply $x$ by $x$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 x^{2} - 70 x \cdot - 1)$

Step 3.1.7.1.8

Multiply $- 1$ by $- 70$ .

$- \frac{3}{4} \cdot (6 x^{5} - 6 x^{4} + 47 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 x^{2} + 70 x)$

Step 3.1.7.2

Simplify terms.

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Step 3.1.7.2.1

Add $- 6 x^{4}$ and $47 x^{4}$ .

$- \frac{3}{4} \cdot (6 x^{5} + 41 x^{4} - 47 x^{3} + 71 x^{3} - 71 x^{2} - 70 x^{2} + 70 x)$

Step 3.1.7.2.2

Add $- 47 x^{3}$ and $71 x^{3}$ .

$- \frac{3}{4} \cdot (6 x^{5} + 41 x^{4} + 24 x^{3} - 71 x^{2} - 70 x^{2} + 70 x)$

Step 3.1.7.2.3

Subtract $70 x^{2}$ from $- 71 x^{2}$ .

$- \frac{3}{4} \cdot (6 x^{5} + 41 x^{4} + 24 x^{3} - 141 x^{2} + 70 x)$

Step 3.1.7.2.4

Apply the distributive property.

$- \frac{3}{4} (6 x^{5}) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8

Simplify.

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Step 3.1.8.1

Cancel the common factor of $2$ .

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Step 3.1.8.1.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 3}{4} (6 x^{5}) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.1.2

Factor $2$ out of $4$ .

$\frac{- 3}{2 (2)} (6 x^{5}) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.1.3

Factor $2$ out of $6 x^{5}$ .

$\frac{- 3}{2 (2)} (2 (3 x^{5})) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.1.4

Cancel the common factor.

$\frac{- 3}{2 \cdot 2} (2 (3 x^{5})) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.1.5

Rewrite the expression.

$\frac{- 3}{2} (3 x^{5}) - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.2

Combine $3$ and $\frac{- 3}{2}$ .

$\frac{3 \cdot - 3}{2} x^{5} - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.3

Multiply $3$ by $- 3$ .

$\frac{- 9}{2} x^{5} - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.4

Combine $\frac{- 9}{2}$ and $x^{5}$ .

$\frac{- 9 x^{5}}{2} - \frac{3}{4} (41 x^{4}) - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.5

Multiply $- \frac{3}{4} (41 x^{4})$ .

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Step 3.1.8.5.1

Multiply $41$ by $- 1$ .

$\frac{- 9 x^{5}}{2} - 41 (\frac{3}{4}) x^{4} - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.5.2

Combine $- 41$ and $\frac{3}{4}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 41 \cdot 3}{4} x^{4} - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.5.3

Multiply $- 41$ by $3$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123}{4} x^{4} - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.5.4

Combine $\frac{- 123}{4}$ and $x^{4}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - \frac{3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.6

Cancel the common factor of $4$ .

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Step 3.1.8.6.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} + \frac{- 3}{4} (24 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.6.2

Factor $4$ out of $24 x^{3}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} + \frac{- 3}{4} (4 (6 x^{3})) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.6.3

Cancel the common factor.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} + \frac{- 3}{4} (4 (6 x^{3})) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.6.4

Rewrite the expression.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 3 (6 x^{3}) - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.7

Multiply $6$ by $- 3$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} - \frac{3}{4} (- 141 x^{2}) - \frac{3}{4} (70 x)$

Step 3.1.8.8

Multiply $- \frac{3}{4} (- 141 x^{2})$ .

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Step 3.1.8.8.1

Multiply $- 141$ by $- 1$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + 141 (\frac{3}{4}) x^{2} - \frac{3}{4} (70 x)$

Step 3.1.8.8.2

Combine $141$ and $\frac{3}{4}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{141 \cdot 3}{4} x^{2} - \frac{3}{4} (70 x)$

Step 3.1.8.8.3

Multiply $141$ by $3$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423}{4} x^{2} - \frac{3}{4} (70 x)$

Step 3.1.8.8.4

Combine $\frac{423}{4}$ and $x^{2}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} - \frac{3}{4} (70 x)$

Step 3.1.8.9

Cancel the common factor of $2$ .

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Step 3.1.8.9.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{4} (70 x)$

Step 3.1.8.9.2

Factor $2$ out of $4$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{2 (2)} (70 x)$

Step 3.1.8.9.3

Factor $2$ out of $70 x$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{2 (2)} (2 (35 x))$

Step 3.1.8.9.4

Cancel the common factor.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{2 \cdot 2} (2 (35 x))$

Step 3.1.8.9.5

Rewrite the expression.

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 3}{2} (35 x)$

Step 3.1.8.10

Combine $35$ and $\frac{- 3}{2}$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{35 \cdot - 3}{2} x$

Step 3.1.8.11

Multiply $35$ by $- 3$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 105}{2} x$

Step 3.1.8.12

Combine $\frac{- 105}{2}$ and $x$ .

$\frac{- 9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 105 x}{2}$

Step 3.1.9

Simplify each term.

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Step 3.1.9.1

Move the negative in front of the fraction.

$- \frac{9 x^{5}}{2} + \frac{- 123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 105 x}{2}$

Step 3.1.9.2

Move the negative in front of the fraction.

$- \frac{9 x^{5}}{2} - \frac{123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} + \frac{- 105 x}{2}$

Step 3.1.9.3

Move the negative in front of the fraction.

$- \frac{9 x^{5}}{2} - \frac{123 x^{4}}{4} - 18 x^{3} + \frac{423 x^{2}}{4} - \frac{105 x}{2}$

Step 3.2

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

$- \frac{9 x^{5}}{2}$

Step 3.3

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

$- \frac{9}{2}$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

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.

Rises to the left and falls to the right

Step 7