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

$f (x) = 2 (x - 1) {(x + 1)}^{2} {(x - 3)}^{3}$

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

Simplify by multiplying through.

Apply the distributive property.

$f (x) = (2 x + 2 \cdot - 1) {(x + 1)}^{2} {(x - 3)}^{3}$

Simplify the expression.

Multiply $2$ by $- 1$ .

$f (x) = (2 x - 2) {(x + 1)}^{2} {(x - 3)}^{3}$

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

$f (x) = (2 x - 2) ((x + 1) (x + 1)) {(x - 3)}^{3}$

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

Apply the distributive property.

$f (x) = (2 x - 2) (x (x + 1) + 1 (x + 1)) {(x - 3)}^{3}$

Apply the distributive property.

$f (x) = (2 x - 2) (x \cdot x + x \cdot 1 + 1 (x + 1)) {(x - 3)}^{3}$

Apply the distributive property.

$f (x) = (2 x - 2) (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) {(x - 3)}^{3}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = (2 x - 2) (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) {(x - 3)}^{3}$

Multiply $x$ by $1$ .

$f (x) = (2 x - 2) (x^{2} + x + 1 x + 1 \cdot 1) {(x - 3)}^{3}$

Multiply $x$ by $1$ .

$f (x) = (2 x - 2) (x^{2} + x + x + 1 \cdot 1) {(x - 3)}^{3}$

Multiply $1$ by $1$ .

$f (x) = (2 x - 2) (x^{2} + x + x + 1) {(x - 3)}^{3}$

Add $x$ and $x$ .

$f (x) = (2 x - 2) (x^{2} + 2 x + 1) {(x - 3)}^{3}$

Expand $(2 x - 2) (x^{2} + 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = (2 x \cdot x^{2} + 2 x (2 x) + 2 x \cdot 1 - 2 x^{2} - 2 (2 x) - 2 \cdot 1) {(x - 3)}^{3}$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$f (x) = (2 (x^{2} x) + 2 x (2 x) + 2 x \cdot 1 - 2 x^{2} - 2 (2 x) - 2 \cdot 1) {(x - 3)}^{3}$

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

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

$f (x) = (2 (x^{2} x) + 2 x (2 x) + 2 x \cdot 1 - 2 x^{2} - 2 (2 x) - 2 \cdot 1) {(x - 3)}^{3}$

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

$f (x) = (2 x^{2 + 1} + 2 x (2 x) + 2 x \cdot 1 - 2 x^{2} - 2 (2 x) - 2 \cdot 1) {(x - 3)}^{3}$

Add $2$ and $1$ .

$f (x) = (2 x^{3} + 2 x (2 x) + 2 x \cdot 1 - 2 x^{2} - 2 (2 x) - 2 \cdot 1) {(x - 3)}^{3}$

Rewrite using the commutative property of multiplication.

$f (x) = (2 x^{3} + 2 \cdot (2 x \cdot x) + 2 x \cdot 1 - 2 x^{2} - 2 (2 x) - 2 \cdot 1) {(x - 3)}^{3}$

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

Move $x$ .

$f (x) = (2 x^{3} + 2 \cdot (2 (x \cdot x)) + 2 x \cdot 1 - 2 x^{2} - 2 (2 x) - 2 \cdot 1) {(x - 3)}^{3}$

Multiply $x$ by $x$ .

$f (x) = (2 x^{3} + 2 \cdot (2 x^{2}) + 2 x \cdot 1 - 2 x^{2} - 2 (2 x) - 2 \cdot 1) {(x - 3)}^{3}$

Multiply $2$ by $2$ .

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

Multiply $2$ by $1$ .

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

Multiply $2$ by $- 2$ .

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

Multiply $- 2$ by $1$ .

$f (x) = (2 x^{3} + 4 x^{2} + 2 x - 2 x^{2} - 4 x - 2) {(x - 3)}^{3}$

Simplify by adding terms.

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

$f (x) = (2 x^{3} + 2 x^{2} + 2 x - 4 x - 2) {(x - 3)}^{3}$

Subtract $4 x$ from $2 x$ .

$f (x) = (2 x^{3} + 2 x^{2} - 2 x - 2) {(x - 3)}^{3}$

Use the Binomial Theorem.

$f (x) = (2 x^{3} + 2 x^{2} - 2 x - 2) (x^{3} + 3 x^{2} \cdot - 3 + 3 x {(- 3)}^{2} + {(- 3)}^{3})$

Simplify each term.

Multiply $- 3$ by $3$ .

$f (x) = (2 x^{3} + 2 x^{2} - 2 x - 2) (x^{3} - 9 x^{2} + 3 x {(- 3)}^{2} + {(- 3)}^{3})$

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

$f (x) = (2 x^{3} + 2 x^{2} - 2 x - 2) (x^{3} - 9 x^{2} + 3 x \cdot 9 + {(- 3)}^{3})$

Multiply $9$ by $3$ .

$f (x) = (2 x^{3} + 2 x^{2} - 2 x - 2) (x^{3} - 9 x^{2} + 27 x + {(- 3)}^{3})$

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

$f (x) = (2 x^{3} + 2 x^{2} - 2 x - 2) (x^{3} - 9 x^{2} + 27 x - 27)$

Expand $(2 x^{3} + 2 x^{2} - 2 x - 2) (x^{3} - 9 x^{2} + 27 x - 27)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = 2 x^{3} x^{3} + 2 x^{3} (- 9 x^{2}) + 2 x^{3} (27 x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Simplify terms.

Simplify each term.

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

Move $x^{3}$ .

$f (x) = 2 (x^{3} x^{3}) + 2 x^{3} (- 9 x^{2}) + 2 x^{3} (27 x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

$f (x) = 2 x^{3 + 3} + 2 x^{3} (- 9 x^{2}) + 2 x^{3} (27 x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Add $3$ and $3$ .

$f (x) = 2 x^{6} + 2 x^{3} (- 9 x^{2}) + 2 x^{3} (27 x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Rewrite using the commutative property of multiplication.

$f (x) = 2 x^{6} + 2 \cdot (- 9 x^{3} x^{2}) + 2 x^{3} (27 x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

Move $x^{2}$ .

$f (x) = 2 x^{6} + 2 \cdot (- 9 (x^{2} x^{3})) + 2 x^{3} (27 x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

$f (x) = 2 x^{6} + 2 \cdot (- 9 x^{2 + 3}) + 2 x^{3} (27 x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Add $2$ and $3$ .

$f (x) = 2 x^{6} + 2 \cdot (- 9 x^{5}) + 2 x^{3} (27 x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $2$ by $- 9$ .

$f (x) = 2 x^{6} - 18 x^{5} + 2 x^{3} (27 x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Rewrite using the commutative property of multiplication.

$f (x) = 2 x^{6} - 18 x^{5} + 2 \cdot (27 x^{3} x) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

Move $x$ .

$f (x) = 2 x^{6} - 18 x^{5} + 2 \cdot (27 (x \cdot x^{3})) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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.

$f (x) = 2 x^{6} - 18 x^{5} + 2 \cdot (27 x^{1 + 3}) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Add $1$ and $3$ .

$f (x) = 2 x^{6} - 18 x^{5} + 2 \cdot (27 x^{4}) + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $2$ by $27$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} + 2 x^{3} \cdot - 27 + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $- 27$ by $2$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{2} x^{3} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

Move $x^{3}$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 (x^{3} x^{2}) + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{3 + 2} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Add $3$ and $2$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} + 2 x^{2} (- 9 x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Rewrite using the commutative property of multiplication.

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} + 2 \cdot (- 9 x^{2} x^{2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

Move $x^{2}$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} + 2 \cdot (- 9 (x^{2} x^{2})) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} + 2 \cdot (- 9 x^{2 + 2}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Add $2$ and $2$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} + 2 \cdot (- 9 x^{4}) + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $2$ by $- 9$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 2 x^{2} (27 x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Rewrite using the commutative property of multiplication.

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 2 \cdot (27 x^{2} x) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

Move $x$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 2 \cdot (27 (x \cdot x^{2})) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

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

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

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 2 \cdot (27 x^{1 + 2}) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Add $1$ and $2$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 2 \cdot (27 x^{3}) + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $2$ by $27$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} + 2 x^{2} \cdot - 27 - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $- 27$ by $2$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x \cdot x^{3} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

Move $x^{3}$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 (x^{3} x) - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

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

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

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{3 + 1} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Add $3$ and $1$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} - 2 x (- 9 x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Rewrite using the commutative property of multiplication.

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} - 2 \cdot (- 9 x \cdot x^{2}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

Move $x^{2}$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} - 2 \cdot (- 9 (x^{2} x)) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

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

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

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} - 2 \cdot (- 9 x^{2 + 1}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Add $2$ and $1$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} - 2 \cdot (- 9 x^{3}) - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $- 2$ by $- 9$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 2 x (27 x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Rewrite using the commutative property of multiplication.

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 2 \cdot (27 x \cdot x) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

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

Move $x$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 2 \cdot (27 (x \cdot x)) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $x$ by $x$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 2 \cdot (27 x^{2}) - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $- 2$ by $27$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} - 2 x \cdot - 27 - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $- 27$ by $- 2$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} + 54 x - 2 x^{3} - 2 (- 9 x^{2}) - 2 (27 x) - 2 \cdot - 27$

Multiply $- 9$ by $- 2$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} + 54 x - 2 x^{3} + 18 x^{2} - 2 (27 x) - 2 \cdot - 27$

Multiply $27$ by $- 2$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} + 54 x - 2 x^{3} + 18 x^{2} - 54 x - 2 \cdot - 27$

Multiply $- 2$ by $- 27$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} + 54 x - 2 x^{3} + 18 x^{2} - 54 x + 54$

Simplify by adding terms.

Combine the opposite terms in $2 x^{6} - 18 x^{5} + 54 x^{4} - 54 x^{3} + 2 x^{5} - 18 x^{4} + 54 x^{3} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} + 54 x - 2 x^{3} + 18 x^{2} - 54 x + 54$ .

Add $- 54 x^{3}$ and $54 x^{3}$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} + 2 x^{5} - 18 x^{4} + 0 - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} + 54 x - 2 x^{3} + 18 x^{2} - 54 x + 54$

Add $2 x^{6} - 18 x^{5} + 54 x^{4} + 2 x^{5} - 18 x^{4}$ and $0$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} + 2 x^{5} - 18 x^{4} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} + 54 x - 2 x^{3} + 18 x^{2} - 54 x + 54$

Subtract $54 x$ from $54 x$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} + 2 x^{5} - 18 x^{4} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} - 2 x^{3} + 18 x^{2} + 0 + 54$

Add $2 x^{6} - 18 x^{5} + 54 x^{4} + 2 x^{5} - 18 x^{4} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} - 2 x^{3} + 18 x^{2}$ and $0$ .

$f (x) = 2 x^{6} - 18 x^{5} + 54 x^{4} + 2 x^{5} - 18 x^{4} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} - 2 x^{3} + 18 x^{2} + 54$

Add $- 18 x^{5}$ and $2 x^{5}$ .

$f (x) = 2 x^{6} - 16 x^{5} + 54 x^{4} - 18 x^{4} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} - 2 x^{3} + 18 x^{2} + 54$

Subtract $18 x^{4}$ from $54 x^{4}$ .

$f (x) = 2 x^{6} - 16 x^{5} + 36 x^{4} - 54 x^{2} - 2 x^{4} + 18 x^{3} - 54 x^{2} - 2 x^{3} + 18 x^{2} + 54$

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

$f (x) = 2 x^{6} - 16 x^{5} + 34 x^{4} - 54 x^{2} + 18 x^{3} - 54 x^{2} - 2 x^{3} + 18 x^{2} + 54$

Subtract $54 x^{2}$ from $- 54 x^{2}$ .

$f (x) = 2 x^{6} - 16 x^{5} + 34 x^{4} + 18 x^{3} - 108 x^{2} - 2 x^{3} + 18 x^{2} + 54$

Subtract $2 x^{3}$ from $18 x^{3}$ .

$f (x) = 2 x^{6} - 16 x^{5} + 34 x^{4} + 16 x^{3} - 108 x^{2} + 18 x^{2} + 54$

Add $- 108 x^{2}$ and $18 x^{2}$ .

$f (x) = 2 x^{6} - 16 x^{5} + 34 x^{4} + 16 x^{3} - 90 x^{2} + 54$

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.

$2 x^{6} \to 6$

$- 16 x^{5} \to 5$

$34 x^{4} \to 4$

$16 x^{3} \to 3$

$- 90 x^{2} \to 2$

$54 \to 0$

The largest exponent is the degree of the polynomial.

$6$

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

$2 x^{6}$

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.

$2 x^{6}$

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

$2$

List the results.

Polynomial Degree: $6$

Leading Term: $2 x^{6}$

Leading Coefficient: $2$

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

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