Find the Behavior (Leading Coefficient Test) f(x)=8x^3(4-x)(7x-6)^4

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

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify terms.

Apply the distributive property.

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

Simplify the expression.

Multiply $4$ by $8$ .

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

Rewrite using the commutative property of multiplication.

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

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $3$ .

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

Multiply $8$ by $- 1$ .

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

Use the Binomial Theorem.

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

Simplify each term.

Apply the product rule to $7 x$ .

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

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

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

Apply the product rule to $7 x$ .

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

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

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

Multiply $343$ by $4$ .

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

Multiply $- 6$ by $1372$ .

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

Apply the product rule to $7 x$ .

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

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

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

Multiply $49$ by $6$ .

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

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

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

Multiply $36$ by $294$ .

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

Multiply $7$ by $4$ .

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

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

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

Multiply $- 216$ by $28$ .

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

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

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

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.

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

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x^{4}$ .

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

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

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

Add $4$ and $3$ .

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

Multiply $32$ by $2401$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{3}$ .

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

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

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

Add $3$ and $3$ .

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

Multiply $32$ by $- 8232$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $3$ .

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

Multiply $32$ by $10584$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $3$ .

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

Multiply $32$ by $- 6048$ .

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

Multiply $1296$ by $32$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{4}$ .

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

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

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

Add $4$ and $4$ .

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

Multiply $- 8$ by $2401$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{3}$ .

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

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

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

Add $3$ and $4$ .

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

Multiply $- 8$ by $- 8232$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $4$ .

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

Multiply $- 8$ by $10584$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $4$ .

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

Multiply $- 8$ by $- 6048$ .

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

Multiply $1296$ by $- 8$ .

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

Simplify by adding terms.

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

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

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

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

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

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

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

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

The largest exponent is the degree of the polynomial.

$8$

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

Even

Identify the leading coefficient.

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

Simplify terms.

Apply the distributive property.

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

Simplify the expression.

Multiply $4$ by $8$ .

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

Rewrite using the commutative property of multiplication.

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

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $3$ .

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

Multiply $8$ by $- 1$ .

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

Use the Binomial Theorem.

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

Simplify each term.

Apply the product rule to $7 x$ .

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

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

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

Apply the product rule to $7 x$ .

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

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

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

Multiply $343$ by $4$ .

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

Multiply $- 6$ by $1372$ .

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

Apply the product rule to $7 x$ .

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

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

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

Multiply $49$ by $6$ .

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

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

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

Multiply $36$ by $294$ .

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

Multiply $7$ by $4$ .

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

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

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

Multiply $- 216$ by $28$ .

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

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

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

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.

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Move $x^{4}$ .

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

Add $4$ and $3$ .

Multiply $32$ by $2401$ .

Rewrite using the commutative property of multiplication.

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

Move $x^{3}$ .

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

Add $3$ and $3$ .

Multiply $32$ by $- 8232$ .

Rewrite using the commutative property of multiplication.

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

Move $x^{2}$ .

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

Add $2$ and $3$ .

Multiply $32$ by $10584$ .

Rewrite using the commutative property of multiplication.

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

Move $x$ .

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

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

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

Add $1$ and $3$ .

Multiply $32$ by $- 6048$ .

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

Multiply $1296$ by $32$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{4}$ .

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

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

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

Add $4$ and $4$ .

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

Multiply $- 8$ by $2401$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{3}$ .

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

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

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

Add $3$ and $4$ .

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

Multiply $- 8$ by $- 8232$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $4$ .

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

Multiply $- 8$ by $10584$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $4$ .

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

Multiply $- 8$ by $- 6048$ .

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

Multiply $1296$ by $- 8$ .

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

Simplify by adding terms.

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

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

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

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

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

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

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

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

Simplify the expression.

Move $41472 x^{3}$ .

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

Move $- 203904 x^{4}$ .

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

Move $387072 x^{5}$ .

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

Move $- 348096 x^{6}$ .

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

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

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

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

$- 19208 x^{8}$

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

$- 19208$

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

Negative

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

Determine the behavior.

Falls to the left and falls to the right

Find the Behavior (Leading Coefficient Test) f(x)=8x^3(4-x)(7x-6)^4

Related Questions

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