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

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

Expand $(- x^{2} - 2 x - 1) (2 x - 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $- 3$ by $- 1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $2$ .

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

Multiply $- 3$ by $- 2$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $- 3$ .

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

Simplify by adding terms.

Subtract $4 x^{2}$ from $3 x^{2}$ .

$f (x) = (- 2 x^{3} - x^{2} + 6 x - 2 x + 3) {(x + 2)}^{2}$

Subtract $2 x$ from $6 x$ .

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

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

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

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $2$ by $2$ .

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

Add $2 x$ and $2 x$ .

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

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

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

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

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

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

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

Add $2$ and $3$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $3$ .

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

Multiply $- 2$ by $4$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 2 x^{3} \cdot 4 - x^{2} x^{2} - x^{2} (4 x) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Multiply $4$ by $- 2$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{2} x^{2} - x^{2} (4 x) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

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

Move $x^{2}$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - (x^{2} x^{2}) - x^{2} (4 x) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

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

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{2 + 2} - x^{2} (4 x) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Add $2$ and $2$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - x^{2} (4 x) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Rewrite using the commutative property of multiplication.

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 1 \cdot (4 x^{2} x) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

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

Move $x$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 1 \cdot (4 (x \cdot x^{2})) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

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

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

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 1 \cdot (4 (x \cdot x^{2})) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

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

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 1 \cdot (4 x^{1 + 2}) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Add $1$ and $2$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 1 \cdot (4 x^{3}) - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Multiply $- 1$ by $4$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Multiply $4$ by $- 1$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

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

Move $x^{2}$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 (x^{2} x) + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

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

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

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 (x^{2} x) + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

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

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{2 + 1} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Add $2$ and $1$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{3} + 4 x (4 x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Rewrite using the commutative property of multiplication.

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{3} + 4 \cdot (4 x \cdot x) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

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

Move $x$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{3} + 4 \cdot (4 (x \cdot x)) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Multiply $x$ by $x$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{3} + 4 \cdot (4 x^{2}) + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Multiply $4$ by $4$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{3} + 16 x^{2} + 4 x \cdot 4 + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Multiply $4$ by $4$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{3} + 16 x^{2} + 16 x + 3 x^{2} + 3 (4 x) + 3 \cdot 4$

Multiply $4$ by $3$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{3} + 16 x^{2} + 16 x + 3 x^{2} + 12 x + 3 \cdot 4$

Multiply $3$ by $4$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{3} + 16 x^{2} + 16 x + 3 x^{2} + 12 x + 12$

Simplify by adding terms.

Combine the opposite terms in $- 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{3} - 4 x^{2} + 4 x^{3} + 16 x^{2} + 16 x + 3 x^{2} + 12 x + 12$ .

Add $- 4 x^{3}$ and $4 x^{3}$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{2} + 0 + 16 x^{2} + 16 x + 3 x^{2} + 12 x + 12$

Add $- 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{2}$ and $0$ .

$f (x) = - 2 x^{5} - 8 x^{4} - 8 x^{3} - x^{4} - 4 x^{2} + 16 x^{2} + 16 x + 3 x^{2} + 12 x + 12$

Subtract $x^{4}$ from $- 8 x^{4}$ .

$f (x) = - 2 x^{5} - 9 x^{4} - 8 x^{3} - 4 x^{2} + 16 x^{2} + 16 x + 3 x^{2} + 12 x + 12$

Add $- 4 x^{2}$ and $16 x^{2}$ .

$f (x) = - 2 x^{5} - 9 x^{4} - 8 x^{3} + 12 x^{2} + 16 x + 3 x^{2} + 12 x + 12$

Add $12 x^{2}$ and $3 x^{2}$ .

$f (x) = - 2 x^{5} - 9 x^{4} - 8 x^{3} + 15 x^{2} + 16 x + 12 x + 12$

Add $16 x$ and $12 x$ .

$f (x) = - 2 x^{5} - 9 x^{4} - 8 x^{3} + 15 x^{2} + 28 x + 12$

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.

$- 2 x^{5} \to 5$

$- 9 x^{4} \to 4$

$- 8 x^{3} \to 3$

$15 x^{2} \to 2$

$28 x \to 1$

$12 \to 0$

The largest exponent is the degree of the polynomial.

$5$

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

$- 2 x^{5}$

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.

$- 2 x^{5}$

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

$- 2$

List the results.

Polynomial Degree: $5$

Leading Term: $- 2 x^{5}$

Leading Coefficient: $- 2$

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

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