Find the Degree, Leading Term, and Leading Coefficient 3x^5(3x+1)^2(2-x)^4

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

Step 1

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

Rewrite ${(3 x + 1)}^{2}$ as $(3 x + 1) (3 x + 1)$ .

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

Expand $(3 x + 1) (3 x + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $3$ .

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

Multiply $3$ by $1$ .

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

Multiply $3 x$ by $1$ .

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

Multiply $1$ by $1$ .

$3 x^{5} (9 x^{2} + 3 x + 3 x + 1) {(2 - x)}^{4}$

Add $3 x$ and $3 x$ .

$3 x^{5} (9 x^{2} + 6 x + 1) {(2 - x)}^{4}$

Apply the distributive property.

$(3 x^{5} (9 x^{2}) + 3 x^{5} (6 x) + 3 x^{5} \cdot 1) {(2 - x)}^{4}$

Simplify.

Rewrite using the commutative property of multiplication.

$(3 \cdot 9 x^{5} x^{2} + 3 x^{5} (6 x) + 3 x^{5} \cdot 1) {(2 - x)}^{4}$

Rewrite using the commutative property of multiplication.

$(3 \cdot 9 x^{5} x^{2} + 3 \cdot 6 x^{5} x + 3 x^{5} \cdot 1) {(2 - x)}^{4}$

Multiply $3$ by $1$ .

$(3 \cdot 9 x^{5} x^{2} + 3 \cdot 6 x^{5} x + 3 x^{5}) {(2 - x)}^{4}$

Simplify each term.

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(3 \cdot 9 (x^{2} x^{5}) + 3 \cdot 6 x^{5} x + 3 x^{5}) {(2 - x)}^{4}$

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

$(3 \cdot 9 x^{2 + 5} + 3 \cdot 6 x^{5} x + 3 x^{5}) {(2 - x)}^{4}$

Add $2$ and $5$ .

$(3 \cdot 9 x^{7} + 3 \cdot 6 x^{5} x + 3 x^{5}) {(2 - x)}^{4}$

Multiply $3$ by $9$ .

$(27 x^{7} + 3 \cdot 6 x^{5} x + 3 x^{5}) {(2 - x)}^{4}$

Multiply $x^{5}$ by $x$ by adding the exponents.

Move $x$ .

$(27 x^{7} + 3 \cdot 6 (x \cdot x^{5}) + 3 x^{5}) {(2 - x)}^{4}$

Multiply $x$ by $x^{5}$ .

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

$(27 x^{7} + 3 \cdot 6 (x^{1} x^{5}) + 3 x^{5}) {(2 - x)}^{4}$

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

$(27 x^{7} + 3 \cdot 6 x^{1 + 5} + 3 x^{5}) {(2 - x)}^{4}$

Add $1$ and $5$ .

$(27 x^{7} + 3 \cdot 6 x^{6} + 3 x^{5}) {(2 - x)}^{4}$

Multiply $3$ by $6$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) {(2 - x)}^{4}$

Use the Binomial Theorem.

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (2^{4} + 4 \cdot 2^{3} (- x) + 6 \cdot 2^{2} {(- x)}^{2} + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

Simplify each term.

Raise $2$ to the power of $4$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 + 4 \cdot 2^{3} (- x) + 6 \cdot 2^{2} {(- x)}^{2} + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

Raise $2$ to the power of $3$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 + 4 \cdot 8 (- x) + 6 \cdot 2^{2} {(- x)}^{2} + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

Multiply $4$ by $8$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 + 32 (- x) + 6 \cdot 2^{2} {(- x)}^{2} + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

Multiply $- 1$ by $32$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 6 \cdot 2^{2} {(- x)}^{2} + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

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

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 6 \cdot 4 {(- x)}^{2} + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

Multiply $6$ by $4$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 {(- x)}^{2} + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

Apply the product rule to $- x$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 ({(- 1)}^{2} x^{2}) + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

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

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 (1 x^{2}) + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

Multiply $x^{2}$ by $1$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 x^{2} + 4 \cdot 2 {(- x)}^{3} + {(- x)}^{4})$

Multiply $4$ by $2$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 x^{2} + 8 {(- x)}^{3} + {(- x)}^{4})$

Apply the product rule to $- x$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 x^{2} + 8 ({(- 1)}^{3} x^{3}) + {(- x)}^{4})$

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

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 x^{2} + 8 (- x^{3}) + {(- x)}^{4})$

Multiply $- 1$ by $8$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 x^{2} - 8 x^{3} + {(- x)}^{4})$

Apply the product rule to $- x$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 x^{2} - 8 x^{3} + {(- 1)}^{4} x^{4})$

Raise $- 1$ to the power of $4$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 x^{2} - 8 x^{3} + 1 x^{4})$

Multiply $x^{4}$ by $1$ .

$(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 x^{2} - 8 x^{3} + x^{4})$

Expand $(27 x^{7} + 18 x^{6} + 3 x^{5}) (16 - 32 x + 24 x^{2} - 8 x^{3} + x^{4})$ by multiplying each term in the first expression by each term in the second expression.

$27 x^{7} \cdot 16 + 27 x^{7} (- 32 x) + 27 x^{7} (24 x^{2}) + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Simplify terms.

Simplify each term.

Multiply $16$ by $27$ .

$432 x^{7} + 27 x^{7} (- 32 x) + 27 x^{7} (24 x^{2}) + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Rewrite using the commutative property of multiplication.

$432 x^{7} + 27 \cdot - 32 x^{7} x + 27 x^{7} (24 x^{2}) + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{7}$ by $x$ by adding the exponents.

Move $x$ .

$432 x^{7} + 27 \cdot - 32 (x \cdot x^{7}) + 27 x^{7} (24 x^{2}) + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x$ by $x^{7}$ .

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

$432 x^{7} + 27 \cdot - 32 (x^{1} x^{7}) + 27 x^{7} (24 x^{2}) + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} + 27 \cdot - 32 x^{1 + 7} + 27 x^{7} (24 x^{2}) + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $1$ and $7$ .

$432 x^{7} + 27 \cdot - 32 x^{8} + 27 x^{7} (24 x^{2}) + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $27$ by $- 32$ .

$432 x^{7} - 864 x^{8} + 27 x^{7} (24 x^{2}) + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Rewrite using the commutative property of multiplication.

$432 x^{7} - 864 x^{8} + 27 \cdot 24 x^{7} x^{2} + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{7}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$432 x^{7} - 864 x^{8} + 27 \cdot 24 (x^{2} x^{7}) + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 27 \cdot 24 x^{2 + 7} + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $2$ and $7$ .

$432 x^{7} - 864 x^{8} + 27 \cdot 24 x^{9} + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $27$ by $24$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} + 27 x^{7} (- 8 x^{3}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Rewrite using the commutative property of multiplication.

$432 x^{7} - 864 x^{8} + 648 x^{9} + 27 \cdot - 8 x^{7} x^{3} + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{7}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} + 27 \cdot - 8 (x^{3} x^{7}) + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} + 27 \cdot - 8 x^{3 + 7} + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $3$ and $7$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} + 27 \cdot - 8 x^{10} + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $27$ by $- 8$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{7} x^{4} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{7}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 (x^{4} x^{7}) + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{4 + 7} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $4$ and $7$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 18 x^{6} \cdot 16 + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $16$ by $18$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} + 18 x^{6} (- 32 x) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Rewrite using the commutative property of multiplication.

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} + 18 \cdot - 32 x^{6} x + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} + 18 \cdot - 32 (x \cdot x^{6}) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x$ by $x^{6}$ .

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} + 18 \cdot - 32 (x^{1} x^{6}) + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} + 18 \cdot - 32 x^{1 + 6} + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $1$ and $6$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} + 18 \cdot - 32 x^{7} + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $18$ by $- 32$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 18 x^{6} (24 x^{2}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Rewrite using the commutative property of multiplication.

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 18 \cdot 24 x^{6} x^{2} + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{6}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 18 \cdot 24 (x^{2} x^{6}) + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 18 \cdot 24 x^{2 + 6} + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $2$ and $6$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 18 \cdot 24 x^{8} + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $18$ by $24$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} + 18 x^{6} (- 8 x^{3}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Rewrite using the commutative property of multiplication.

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} + 18 \cdot - 8 x^{6} x^{3} + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{6}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} + 18 \cdot - 8 (x^{3} x^{6}) + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} + 18 \cdot - 8 x^{3 + 6} + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $3$ and $6$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} + 18 \cdot - 8 x^{9} + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $18$ by $- 8$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{6} x^{4} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{6}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 (x^{4} x^{6}) + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{4 + 6} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $4$ and $6$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 3 x^{5} \cdot 16 + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $16$ by $3$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} + 3 x^{5} (- 32 x) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Rewrite using the commutative property of multiplication.

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} + 3 \cdot - 32 x^{5} x + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{5}$ by $x$ by adding the exponents.

Move $x$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} + 3 \cdot - 32 (x \cdot x^{5}) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x$ by $x^{5}$ .

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} + 3 \cdot - 32 (x^{1} x^{5}) + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} + 3 \cdot - 32 x^{1 + 5} + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $1$ and $5$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} + 3 \cdot - 32 x^{6} + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $3$ by $- 32$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 3 x^{5} (24 x^{2}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Rewrite using the commutative property of multiplication.

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 3 \cdot 24 x^{5} x^{2} + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 3 \cdot 24 (x^{2} x^{5}) + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 3 \cdot 24 x^{2 + 5} + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Add $2$ and $5$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 3 \cdot 24 x^{7} + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Multiply $3$ by $24$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} + 3 x^{5} (- 8 x^{3}) + 3 x^{5} x^{4}$

Rewrite using the commutative property of multiplication.

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} + 3 \cdot - 8 x^{5} x^{3} + 3 x^{5} x^{4}$

Multiply $x^{5}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} + 3 \cdot - 8 (x^{3} x^{5}) + 3 x^{5} x^{4}$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} + 3 \cdot - 8 x^{3 + 5} + 3 x^{5} x^{4}$

Add $3$ and $5$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} + 3 \cdot - 8 x^{8} + 3 x^{5} x^{4}$

Multiply $3$ by $- 8$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} - 24 x^{8} + 3 x^{5} x^{4}$

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} - 24 x^{8} + 3 (x^{4} x^{5})$

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

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} - 24 x^{8} + 3 x^{4 + 5}$

Add $4$ and $5$ .

$432 x^{7} - 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 576 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} - 24 x^{8} + 3 x^{9}$

Simplify by adding terms.

Subtract $576 x^{7}$ from $432 x^{7}$ .

$- 864 x^{8} + 648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 144 x^{7} + 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} - 24 x^{8} + 3 x^{9}$

Add $- 864 x^{8}$ and $432 x^{8}$ .

$648 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 144 x^{7} - 432 x^{8} - 144 x^{9} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} - 24 x^{8} + 3 x^{9}$

Subtract $144 x^{9}$ from $648 x^{9}$ .

$504 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 144 x^{7} - 432 x^{8} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} - 24 x^{8} + 3 x^{9}$

Add $504 x^{9}$ and $3 x^{9}$ .

$507 x^{9} - 216 x^{10} + 27 x^{11} + 288 x^{6} - 144 x^{7} - 432 x^{8} + 18 x^{10} + 48 x^{5} - 96 x^{6} + 72 x^{7} - 24 x^{8}$

Add $- 216 x^{10}$ and $18 x^{10}$ .

$507 x^{9} - 198 x^{10} + 27 x^{11} + 288 x^{6} - 144 x^{7} - 432 x^{8} + 48 x^{5} - 96 x^{6} + 72 x^{7} - 24 x^{8}$

Subtract $96 x^{6}$ from $288 x^{6}$ .

$507 x^{9} - 198 x^{10} + 27 x^{11} - 144 x^{7} - 432 x^{8} + 48 x^{5} + 192 x^{6} + 72 x^{7} - 24 x^{8}$

Add $- 144 x^{7}$ and $72 x^{7}$ .

$507 x^{9} - 198 x^{10} + 27 x^{11} - 72 x^{7} - 432 x^{8} + 48 x^{5} + 192 x^{6} - 24 x^{8}$

Subtract $24 x^{8}$ from $- 432 x^{8}$ .

$507 x^{9} - 198 x^{10} + 27 x^{11} - 72 x^{7} - 456 x^{8} + 48 x^{5} + 192 x^{6}$

Simplify the expression.

Move $48 x^{5}$ .

$507 x^{9} - 198 x^{10} + 27 x^{11} - 72 x^{7} - 456 x^{8} + 192 x^{6} + 48 x^{5}$

Move $- 72 x^{7}$ .

$507 x^{9} - 198 x^{10} + 27 x^{11} - 456 x^{8} - 72 x^{7} + 192 x^{6} + 48 x^{5}$

Move $507 x^{9}$ .

$- 198 x^{10} + 27 x^{11} + 507 x^{9} - 456 x^{8} - 72 x^{7} + 192 x^{6} + 48 x^{5}$

Reorder $- 198 x^{10}$ and $27 x^{11}$ .

$27 x^{11} - 198 x^{10} + 507 x^{9} - 456 x^{8} - 72 x^{7} + 192 x^{6} + 48 x^{5}$

Step 2

The degree of a polynomial is the highest degree of its terms.

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$27 x^{11} \to 11$

$- 198 x^{10} \to 10$

$507 x^{9} \to 9$

$- 456 x^{8} \to 8$

$- 72 x^{7} \to 7$

$192 x^{6} \to 6$

$48 x^{5} \to 5$

The largest exponent is the degree of the polynomial.

$11$

Step 3

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

$27 x^{11}$

Step 4

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

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

$27 x^{11}$

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

$27$

Step 5

List the results.

Polynomial Degree: $11$

Leading Term: $27 x^{11}$

Leading Coefficient: $27$

Find the Degree, Leading Term, and Leading Coefficient 3x^5(3x+1)^2(2-x)^4

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