Find the Degree, Leading Term, and Leading Coefficient v(t)=-t(t+2)^2(t-3)

$v (t) = - t {(t + 2)}^{2} (t - 3)$

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

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

$v (t) = - t ((t + 2) (t + 2)) (t - 3)$

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

Apply the distributive property.

$v (t) = - t (t (t + 2) + 2 (t + 2)) (t - 3)$

Apply the distributive property.

$v (t) = - t (t \cdot t + t \cdot 2 + 2 (t + 2)) (t - 3)$

Apply the distributive property.

$v (t) = - t (t \cdot t + t \cdot 2 + 2 t + 2 \cdot 2) (t - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $t$ by $t$ .

$v (t) = - t (t^{2} + t \cdot 2 + 2 t + 2 \cdot 2) (t - 3)$

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

$v (t) = - t (t^{2} + 2 \cdot t + 2 t + 2 \cdot 2) (t - 3)$

Multiply $2$ by $2$ .

$v (t) = - t (t^{2} + 2 t + 2 t + 4) (t - 3)$

Add $2 t$ and $2 t$ .

$v (t) = - t (t^{2} + 4 t + 4) (t - 3)$

Apply the distributive property.

$v (t) = (- t \cdot t^{2} - t (4 t) - t \cdot 4) (t - 3)$

Simplify.

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

Move $t^{2}$ .

$v (t) = (- (t^{2} t) - t (4 t) - t \cdot 4) (t - 3)$

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

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

$v (t) = (- (t^{2} t) - t (4 t) - t \cdot 4) (t - 3)$

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

$v (t) = (- t^{2 + 1} - t (4 t) - t \cdot 4) (t - 3)$

Add $2$ and $1$ .

$v (t) = (- t^{3} - t (4 t) - t \cdot 4) (t - 3)$

Rewrite using the commutative property of multiplication.

$v (t) = (- t^{3} - 1 \cdot (4 t \cdot t) - t \cdot 4) (t - 3)$

Multiply $4$ by $- 1$ .

$v (t) = (- t^{3} - 1 \cdot (4 t \cdot t) - 4 t) (t - 3)$

Simplify each term.

Multiply $t$ by $t$ by adding the exponents.

Move $t$ .

$v (t) = (- t^{3} - 1 \cdot (4 (t \cdot t)) - 4 t) (t - 3)$

Multiply $t$ by $t$ .

$v (t) = (- t^{3} - 1 \cdot (4 t^{2}) - 4 t) (t - 3)$

Multiply $- 1$ by $4$ .

$v (t) = (- t^{3} - 4 t^{2} - 4 t) (t - 3)$

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

$v (t) = - t^{3} t - t^{3} \cdot - 3 - 4 t^{2} t - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

Simplify terms.

Simplify each term.

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

Move $t$ .

$v (t) = - (t \cdot t^{3}) - t^{3} \cdot - 3 - 4 t^{2} t - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

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

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

$v (t) = - (t \cdot t^{3}) - t^{3} \cdot - 3 - 4 t^{2} t - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

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

$v (t) = - t^{1 + 3} - t^{3} \cdot - 3 - 4 t^{2} t - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

Add $1$ and $3$ .

$v (t) = - t^{4} - t^{3} \cdot - 3 - 4 t^{2} t - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

Multiply $- 3$ by $- 1$ .

$v (t) = - t^{4} + 3 t^{3} - 4 t^{2} t - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

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

Move $t$ .

$v (t) = - t^{4} + 3 t^{3} - 4 (t \cdot t^{2}) - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

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

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

$v (t) = - t^{4} + 3 t^{3} - 4 (t \cdot t^{2}) - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

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

$v (t) = - t^{4} + 3 t^{3} - 4 t^{1 + 2} - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

Add $1$ and $2$ .

$v (t) = - t^{4} + 3 t^{3} - 4 t^{3} - 4 t^{2} \cdot - 3 - 4 t \cdot t - 4 t \cdot - 3$

Multiply $- 3$ by $- 4$ .

$v (t) = - t^{4} + 3 t^{3} - 4 t^{3} + 12 t^{2} - 4 t \cdot t - 4 t \cdot - 3$

Multiply $t$ by $t$ by adding the exponents.

Move $t$ .

$v (t) = - t^{4} + 3 t^{3} - 4 t^{3} + 12 t^{2} - 4 (t \cdot t) - 4 t \cdot - 3$

Multiply $t$ by $t$ .

$v (t) = - t^{4} + 3 t^{3} - 4 t^{3} + 12 t^{2} - 4 t^{2} - 4 t \cdot - 3$

Multiply $- 3$ by $- 4$ .

$v (t) = - t^{4} + 3 t^{3} - 4 t^{3} + 12 t^{2} - 4 t^{2} + 12 t$

Simplify by adding terms.

Subtract $4 t^{3}$ from $3 t^{3}$ .

$v (t) = - t^{4} - t^{3} + 12 t^{2} - 4 t^{2} + 12 t$

Subtract $4 t^{2}$ from $12 t^{2}$ .

$v (t) = - t^{4} - t^{3} + 8 t^{2} + 12 t$

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.

$- t^{4} \to 4$

$- t^{3} \to 3$

$8 t^{2} \to 2$

$12 t \to 1$

The largest exponent is the degree of the polynomial.

$4$

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

$- t^{4}$

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.

$- t^{4}$

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

$- 1$

List the results.

Polynomial Degree: $4$

Leading Term: $- t^{4}$

Leading Coefficient: $- 1$

Find the Degree, Leading Term, and Leading Coefficient v(t)=-t(t+2)^2(t-3)

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