Algebra Examples

$f (x) = 8 x^{3} (4 - x) {(7 x - 6)}^{4}$

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.

$(8 x^{3} \cdot 4 + 8 x^{3} (- x)) {(7 x - 6)}^{4}$

Step 1.1.1.2

Simplify the expression.

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Step 1.1.1.2.1

Multiply $4$ by $8$ .

$(32 x^{3} + 8 x^{3} (- x)) {(7 x - 6)}^{4}$

Step 1.1.1.2.2

Rewrite using the commutative property of multiplication.

$(32 x^{3} + 8 \cdot - 1 x^{3} x) {(7 x - 6)}^{4}$

Step 1.1.1.3

Simplify each term.

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Step 1.1.1.3.1

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

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Step 1.1.1.3.1.1

Move $x$ .

$(32 x^{3} + 8 \cdot - 1 (x \cdot x^{3})) {(7 x - 6)}^{4}$

Step 1.1.1.3.1.2

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

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Step 1.1.1.3.1.2.1

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

$(32 x^{3} + 8 \cdot - 1 (x^{1} x^{3})) {(7 x - 6)}^{4}$

Step 1.1.1.3.1.2.2

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

$(32 x^{3} + 8 \cdot - 1 x^{1 + 3}) {(7 x - 6)}^{4}$

Step 1.1.1.3.1.3

Add $1$ and $3$ .

$(32 x^{3} + 8 \cdot - 1 x^{4}) {(7 x - 6)}^{4}$

Step 1.1.1.3.2

Multiply $8$ by $- 1$ .

$(32 x^{3} - 8 x^{4}) {(7 x - 6)}^{4}$

Step 1.1.2

Use the Binomial Theorem.

$(32 x^{3} - 8 x^{4}) ({(7 x)}^{4} + 4 {(7 x)}^{3} \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

Apply the product rule to $7 x$ .

$(32 x^{3} - 8 x^{4}) (7^{4} x^{4} + 4 {(7 x)}^{3} \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.2

Raise $7$ to the power of $4$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} + 4 {(7 x)}^{3} \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.3

Apply the product rule to $7 x$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} + 4 (7^{3} x^{3}) \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.4

Raise $7$ to the power of $3$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} + 4 (343 x^{3}) \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.5

Multiply $343$ by $4$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} + 1372 x^{3} \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.6

Multiply $- 6$ by $1372$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.7

Apply the product rule to $7 x$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 6 (7^{2} x^{2}) {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.8

Raise $7$ to the power of $2$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 6 (49 x^{2}) {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.9

Multiply $49$ by $6$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 294 x^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.10

Raise $- 6$ to the power of $2$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 294 x^{2} \cdot 36 + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.11

Multiply $36$ by $294$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.12

Multiply $7$ by $4$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} + 28 x {(- 6)}^{3} + {(- 6)}^{4})$

Step 1.1.3.13

Raise $- 6$ to the power of $3$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} + 28 x \cdot - 216 + {(- 6)}^{4})$

Step 1.1.3.14

Multiply $- 216$ by $28$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} - 6048 x + {(- 6)}^{4})$

Step 1.1.3.15

Raise $- 6$ to the power of $4$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} - 6048 x + 1296)$

Step 1.1.4

Expand $(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} - 6048 x + 1296)$ by multiplying each term in the first expression by each term in the second expression.

$32 x^{3} (2401 x^{4}) + 32 x^{3} (- 8232 x^{3}) + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

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

Rewrite using the commutative property of multiplication.

$32 \cdot 2401 x^{3} x^{4} + 32 x^{3} (- 8232 x^{3}) + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.2

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 1.1.5.1.2.1

Move $x^{4}$ .

$32 \cdot 2401 (x^{4} x^{3}) + 32 x^{3} (- 8232 x^{3}) + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.2.2

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

$32 \cdot 2401 x^{4 + 3} + 32 x^{3} (- 8232 x^{3}) + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.2.3

Add $4$ and $3$ .

$32 \cdot 2401 x^{7} + 32 x^{3} (- 8232 x^{3}) + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.3

Multiply $32$ by $2401$ .

$76832 x^{7} + 32 x^{3} (- 8232 x^{3}) + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.4

Rewrite using the commutative property of multiplication.

$76832 x^{7} + 32 \cdot - 8232 x^{3} x^{3} + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.5

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

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Step 1.1.5.1.5.1

Move $x^{3}$ .

$76832 x^{7} + 32 \cdot - 8232 (x^{3} x^{3}) + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.5.2

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

$76832 x^{7} + 32 \cdot - 8232 x^{3 + 3} + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.5.3

Add $3$ and $3$ .

$76832 x^{7} + 32 \cdot - 8232 x^{6} + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.6

Multiply $32$ by $- 8232$ .

$76832 x^{7} - 263424 x^{6} + 32 x^{3} (10584 x^{2}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.7

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 32 \cdot 10584 x^{3} x^{2} + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.8

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

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Step 1.1.5.1.8.1

Move $x^{2}$ .

$76832 x^{7} - 263424 x^{6} + 32 \cdot 10584 (x^{2} x^{3}) + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.8.2

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

$76832 x^{7} - 263424 x^{6} + 32 \cdot 10584 x^{2 + 3} + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.8.3

Add $2$ and $3$ .

$76832 x^{7} - 263424 x^{6} + 32 \cdot 10584 x^{5} + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.9

Multiply $32$ by $10584$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} + 32 x^{3} (- 6048 x) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.10

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} + 32 \cdot - 6048 x^{3} x + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.11

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

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Step 1.1.5.1.11.1

Move $x$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} + 32 \cdot - 6048 (x \cdot x^{3}) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.11.2

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

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Step 1.1.5.1.11.2.1

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} + 32 \cdot - 6048 (x^{1} x^{3}) + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.11.2.2

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} + 32 \cdot - 6048 x^{1 + 3} + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.11.3

Add $1$ and $3$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} + 32 \cdot - 6048 x^{4} + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.12

Multiply $32$ by $- 6048$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.13

Multiply $1296$ by $32$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.14

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 \cdot 2401 x^{4} x^{4} - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.15

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

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Step 1.1.5.1.15.1

Move $x^{4}$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 \cdot 2401 (x^{4} x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.15.2

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 \cdot 2401 x^{4 + 4} - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.15.3

Add $4$ and $4$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 \cdot 2401 x^{8} - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.16

Multiply $- 8$ by $2401$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.17

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 \cdot - 8232 x^{4} x^{3} - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.18

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 1.1.5.1.18.1

Move $x^{3}$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 \cdot - 8232 (x^{3} x^{4}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.18.2

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 \cdot - 8232 x^{3 + 4} - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.18.3

Add $3$ and $4$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 \cdot - 8232 x^{7} - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.19

Multiply $- 8$ by $- 8232$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.20

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 \cdot 10584 x^{4} x^{2} - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.21

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 1.1.5.1.21.1

Move $x^{2}$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 \cdot 10584 (x^{2} x^{4}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.21.2

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 \cdot 10584 x^{2 + 4} - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.21.3

Add $2$ and $4$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 \cdot 10584 x^{6} - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.22

Multiply $- 8$ by $10584$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.23

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 x^{4} x - 8 x^{4} \cdot 1296$

Step 1.1.5.1.24

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

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Step 1.1.5.1.24.1

Move $x$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 (x \cdot x^{4}) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.24.2

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

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Step 1.1.5.1.24.2.1

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 (x^{1} x^{4}) - 8 x^{4} \cdot 1296$

Step 1.1.5.1.24.2.2

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 x^{1 + 4} - 8 x^{4} \cdot 1296$

Step 1.1.5.1.24.3

Add $1$ and $4$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 x^{5} - 8 x^{4} \cdot 1296$

Step 1.1.5.1.25

Multiply $- 8$ by $- 6048$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} + 48384 x^{5} - 8 x^{4} \cdot 1296$

Step 1.1.5.1.26

Multiply $1296$ by $- 8$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} + 48384 x^{5} - 10368 x^{4}$

Step 1.1.5.2

Simplify by adding terms.

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Step 1.1.5.2.1

Add $76832 x^{7}$ and $65856 x^{7}$ .

$142688 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 84672 x^{6} + 48384 x^{5} - 10368 x^{4}$

Step 1.1.5.2.2

Subtract $84672 x^{6}$ from $- 263424 x^{6}$ .

$142688 x^{7} - 348096 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 48384 x^{5} - 10368 x^{4}$

Step 1.1.5.2.3

Add $338688 x^{5}$ and $48384 x^{5}$ .

$142688 x^{7} - 348096 x^{6} + 387072 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 10368 x^{4}$

Step 1.1.5.2.4

Subtract $10368 x^{4}$ from $- 193536 x^{4}$ .

$142688 x^{7} - 348096 x^{6} + 387072 x^{5} - 203904 x^{4} + 41472 x^{3} - 19208 x^{8}$

Step 1.2

The largest exponent is the degree of the polynomial.

$8$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

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.

$(8 x^{3} \cdot 4 + 8 x^{3} (- x)) {(7 x - 6)}^{4}$

Step 3.1.1.2

Simplify the expression.

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Step 3.1.1.2.1

Multiply $4$ by $8$ .

$(32 x^{3} + 8 x^{3} (- x)) {(7 x - 6)}^{4}$

Step 3.1.1.2.2

Rewrite using the commutative property of multiplication.

$(32 x^{3} + 8 \cdot - 1 x^{3} x) {(7 x - 6)}^{4}$

Step 3.1.1.3

Simplify each term.

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Step 3.1.1.3.1

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

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Step 3.1.1.3.1.1

Move $x$ .

$(32 x^{3} + 8 \cdot - 1 (x \cdot x^{3})) {(7 x - 6)}^{4}$

Step 3.1.1.3.1.2

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

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Step 3.1.1.3.1.2.1

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

$(32 x^{3} + 8 \cdot - 1 (x^{1} x^{3})) {(7 x - 6)}^{4}$

Step 3.1.1.3.1.2.2

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

$(32 x^{3} + 8 \cdot - 1 x^{1 + 3}) {(7 x - 6)}^{4}$

Step 3.1.1.3.1.3

Add $1$ and $3$ .

$(32 x^{3} + 8 \cdot - 1 x^{4}) {(7 x - 6)}^{4}$

Step 3.1.1.3.2

Multiply $8$ by $- 1$ .

$(32 x^{3} - 8 x^{4}) {(7 x - 6)}^{4}$

Step 3.1.2

Use the Binomial Theorem.

$(32 x^{3} - 8 x^{4}) ({(7 x)}^{4} + 4 {(7 x)}^{3} \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

Apply the product rule to $7 x$ .

$(32 x^{3} - 8 x^{4}) (7^{4} x^{4} + 4 {(7 x)}^{3} \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.2

Raise $7$ to the power of $4$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} + 4 {(7 x)}^{3} \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.3

Apply the product rule to $7 x$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} + 4 (7^{3} x^{3}) \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.4

Raise $7$ to the power of $3$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} + 4 (343 x^{3}) \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.5

Multiply $343$ by $4$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} + 1372 x^{3} \cdot - 6 + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.6

Multiply $- 6$ by $1372$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 6 {(7 x)}^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.7

Apply the product rule to $7 x$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 6 (7^{2} x^{2}) {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.8

Raise $7$ to the power of $2$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 6 (49 x^{2}) {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.9

Multiply $49$ by $6$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 294 x^{2} {(- 6)}^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.10

Raise $- 6$ to the power of $2$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 294 x^{2} \cdot 36 + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.11

Multiply $36$ by $294$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} + 4 (7 x) {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.12

Multiply $7$ by $4$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} + 28 x {(- 6)}^{3} + {(- 6)}^{4})$

Step 3.1.3.13

Raise $- 6$ to the power of $3$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} + 28 x \cdot - 216 + {(- 6)}^{4})$

Step 3.1.3.14

Multiply $- 216$ by $28$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} - 6048 x + {(- 6)}^{4})$

Step 3.1.3.15

Raise $- 6$ to the power of $4$ .

$(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} - 6048 x + 1296)$

Step 3.1.4

Expand $(32 x^{3} - 8 x^{4}) (2401 x^{4} - 8232 x^{3} + 10584 x^{2} - 6048 x + 1296)$ by multiplying each term in the first expression by each term in the second expression.

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

Rewrite using the commutative property of multiplication.

Step 3.1.5.1.2

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 3.1.5.1.2.1

Move $x^{4}$ .

Step 3.1.5.1.2.2

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

Step 3.1.5.1.2.3

Add $4$ and $3$ .

Step 3.1.5.1.3

Multiply $32$ by $2401$ .

Step 3.1.5.1.4

Rewrite using the commutative property of multiplication.

Step 3.1.5.1.5

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

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Step 3.1.5.1.5.1

Move $x^{3}$ .

Step 3.1.5.1.5.2

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

Step 3.1.5.1.5.3

Add $3$ and $3$ .

Step 3.1.5.1.6

Multiply $32$ by $- 8232$ .

Step 3.1.5.1.7

Rewrite using the commutative property of multiplication.

Step 3.1.5.1.8

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

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Step 3.1.5.1.8.1

Move $x^{2}$ .

Step 3.1.5.1.8.2

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

Step 3.1.5.1.8.3

Add $2$ and $3$ .

Step 3.1.5.1.9

Multiply $32$ by $10584$ .

Step 3.1.5.1.10

Rewrite using the commutative property of multiplication.

Step 3.1.5.1.11

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

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Step 3.1.5.1.11.1

Move $x$ .

Step 3.1.5.1.11.2

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

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Step 3.1.5.1.11.2.1

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

Step 3.1.5.1.11.2.2

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

Step 3.1.5.1.11.3

Add $1$ and $3$ .

Step 3.1.5.1.12

Multiply $32$ by $- 6048$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 32 x^{3} \cdot 1296 - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.13

Multiply $1296$ by $32$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 x^{4} (2401 x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.14

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 \cdot 2401 x^{4} x^{4} - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.15

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

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Step 3.1.5.1.15.1

Move $x^{4}$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 \cdot 2401 (x^{4} x^{4}) - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.15.2

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 \cdot 2401 x^{4 + 4} - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.15.3

Add $4$ and $4$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 8 \cdot 2401 x^{8} - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.16

Multiply $- 8$ by $2401$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 x^{4} (- 8232 x^{3}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.17

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 \cdot - 8232 x^{4} x^{3} - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.18

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 3.1.5.1.18.1

Move $x^{3}$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 \cdot - 8232 (x^{3} x^{4}) - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.18.2

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 \cdot - 8232 x^{3 + 4} - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.18.3

Add $3$ and $4$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 8 \cdot - 8232 x^{7} - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.19

Multiply $- 8$ by $- 8232$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 x^{4} (10584 x^{2}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.20

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 \cdot 10584 x^{4} x^{2} - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.21

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 3.1.5.1.21.1

Move $x^{2}$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 \cdot 10584 (x^{2} x^{4}) - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.21.2

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 \cdot 10584 x^{2 + 4} - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.21.3

Add $2$ and $4$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 8 \cdot 10584 x^{6} - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.22

Multiply $- 8$ by $10584$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 x^{4} (- 6048 x) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.23

Rewrite using the commutative property of multiplication.

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 x^{4} x - 8 x^{4} \cdot 1296$

Step 3.1.5.1.24

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

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Step 3.1.5.1.24.1

Move $x$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 (x \cdot x^{4}) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.24.2

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

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Step 3.1.5.1.24.2.1

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 (x^{1} x^{4}) - 8 x^{4} \cdot 1296$

Step 3.1.5.1.24.2.2

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

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 x^{1 + 4} - 8 x^{4} \cdot 1296$

Step 3.1.5.1.24.3

Add $1$ and $4$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} - 8 \cdot - 6048 x^{5} - 8 x^{4} \cdot 1296$

Step 3.1.5.1.25

Multiply $- 8$ by $- 6048$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} + 48384 x^{5} - 8 x^{4} \cdot 1296$

Step 3.1.5.1.26

Multiply $1296$ by $- 8$ .

$76832 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 65856 x^{7} - 84672 x^{6} + 48384 x^{5} - 10368 x^{4}$

Step 3.1.5.2

Simplify by adding terms.

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Step 3.1.5.2.1

Add $76832 x^{7}$ and $65856 x^{7}$ .

$142688 x^{7} - 263424 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 84672 x^{6} + 48384 x^{5} - 10368 x^{4}$

Step 3.1.5.2.2

Subtract $84672 x^{6}$ from $- 263424 x^{6}$ .

$142688 x^{7} - 348096 x^{6} + 338688 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} + 48384 x^{5} - 10368 x^{4}$

Step 3.1.5.2.3

Add $338688 x^{5}$ and $48384 x^{5}$ .

$142688 x^{7} - 348096 x^{6} + 387072 x^{5} - 193536 x^{4} + 41472 x^{3} - 19208 x^{8} - 10368 x^{4}$

Step 3.1.5.2.4

Subtract $10368 x^{4}$ from $- 193536 x^{4}$ .

$142688 x^{7} - 348096 x^{6} + 387072 x^{5} - 203904 x^{4} + 41472 x^{3} - 19208 x^{8}$

Step 3.1.5.2.5

Simplify the expression.

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Step 3.1.5.2.5.1

Move $41472 x^{3}$ .

$142688 x^{7} - 348096 x^{6} + 387072 x^{5} - 203904 x^{4} - 19208 x^{8} + 41472 x^{3}$

Step 3.1.5.2.5.2

Move $- 203904 x^{4}$ .

$142688 x^{7} - 348096 x^{6} + 387072 x^{5} - 19208 x^{8} - 203904 x^{4} + 41472 x^{3}$

Step 3.1.5.2.5.3

Move $387072 x^{5}$ .

$142688 x^{7} - 348096 x^{6} - 19208 x^{8} + 387072 x^{5} - 203904 x^{4} + 41472 x^{3}$

Step 3.1.5.2.5.4

Move $- 348096 x^{6}$ .

$142688 x^{7} - 19208 x^{8} - 348096 x^{6} + 387072 x^{5} - 203904 x^{4} + 41472 x^{3}$

Step 3.1.5.2.5.5

Reorder $142688 x^{7}$ and $- 19208 x^{8}$ .

$- 19208 x^{8} + 142688 x^{7} - 348096 x^{6} + 387072 x^{5} - 203904 x^{4} + 41472 x^{3}$

Step 3.2

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

$- 19208 x^{8}$

Step 3.3

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

$- 19208$

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.

Falls to the left and falls to the right

Step 7