Find the Degree, Leading Term, and Leading Coefficient q(t)=-4t(2-t)(t+1)^3

$q (t) = - 4 t (2 - t) {(t + 1)}^{3}$

Step 1

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

Use the Binomial Theorem.

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

Simplify terms.

Simplify each term.

Multiply $3$ by $1$ .

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

One to any power is one.

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

Multiply $3$ by $1$ .

$q (t) = - 4 t (2 - t) (t^{3} + 3 t^{2} + 3 t + 1^{3})$

One to any power is one.

$q (t) = - 4 t (2 - t) (t^{3} + 3 t^{2} + 3 t + 1)$

Apply the distributive property.

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

Simplify.

Multiply $3$ by $- 4$ .

$q (t) = - 4 t (2 - t) t^{3} - 12 t (2 - t) t^{2} - 4 t (2 - t) (3 t) - 4 t (2 - t) \cdot 1$

Multiply $3$ by $- 4$ .

$q (t) = - 4 t (2 - t) t^{3} - 12 t (2 - t) t^{2} - 12 t (2 - t) t - 4 t (2 - t) \cdot 1$

Multiply $- 4$ by $1$ .

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

Reorder factors in $- 4 t (2 - t) t^{3} - 12 t (2 - t) t^{2} - 12 t (2 - t) t - 4 t (2 - t)$ .

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

Reorder $2$ and $- t$ .

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

Reorder $2$ and $- t$ .

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

Reorder $2$ and $- t$ .

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

Reorder $2$ and $- t$ .

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

Step 2

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

Simplify and reorder the polynomial.

Simplify each term.

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

Move $t$ .

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

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

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

$- 4 (t^{1} t^{3}) (- t + 2) - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

$- 4 t^{1 + 3} (- t + 2) - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Add $1$ and $3$ .

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

Apply the distributive property.

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

Rewrite using the commutative property of multiplication.

$- 4 \cdot - 1 t^{4} t - 4 t^{4} \cdot 2 - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Multiply $2$ by $- 4$ .

$- 4 \cdot - 1 t^{4} t - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Simplify each term.

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

Move $t$ .

$- 4 \cdot - 1 (t \cdot t^{4}) - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

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

$- 4 \cdot - 1 (t^{1} t^{4}) - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

$- 4 \cdot - 1 t^{1 + 4} - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Add $1$ and $4$ .

$- 4 \cdot - 1 t^{5} - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Multiply $- 4$ by $- 1$ .

$4 t^{5} - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

Move $t$ .

$4 t^{5} - 8 t^{4} - 12 (t \cdot t^{2}) (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

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

$4 t^{5} - 8 t^{4} - 12 (t^{1} t^{2}) (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

$4 t^{5} - 8 t^{4} - 12 t^{1 + 2} (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Add $1$ and $2$ .

$4 t^{5} - 8 t^{4} - 12 t^{3} (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Apply the distributive property.

$4 t^{5} - 8 t^{4} - 12 t^{3} (- t) - 12 t^{3} \cdot 2 - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Rewrite using the commutative property of multiplication.

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 t^{3} t - 12 t^{3} \cdot 2 - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Multiply $2$ by $- 12$ .

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 t^{3} t - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Simplify each term.

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

Move $t$ .

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 (t \cdot t^{3}) - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

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

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 (t^{1} t^{3}) - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 t^{1 + 3} - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Add $1$ and $3$ .

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 t^{4} - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Multiply $- 12$ by $- 1$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

Move $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 (t \cdot t) (- t + 2) - 4 t (- t + 2)$

Multiply $t$ by $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 t^{2} (- t + 2) - 4 t (- t + 2)$

Apply the distributive property.

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 t^{2} (- t) - 12 t^{2} \cdot 2 - 4 t (- t + 2)$

Rewrite using the commutative property of multiplication.

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 t^{2} t - 12 t^{2} \cdot 2 - 4 t (- t + 2)$

Multiply $2$ by $- 12$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 t^{2} t - 24 t^{2} - 4 t (- t + 2)$

Simplify each term.

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

Move $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 (t \cdot t^{2}) - 24 t^{2} - 4 t (- t + 2)$

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

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

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 (t^{1} t^{2}) - 24 t^{2} - 4 t (- t + 2)$

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

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 t^{1 + 2} - 24 t^{2} - 4 t (- t + 2)$

Add $1$ and $2$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 t^{3} - 24 t^{2} - 4 t (- t + 2)$

Multiply $- 12$ by $- 1$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 t (- t + 2)$

Apply the distributive property.

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 t (- t) - 4 t \cdot 2$

Rewrite using the commutative property of multiplication.

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 \cdot - 1 t \cdot t - 4 t \cdot 2$

Multiply $2$ by $- 4$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 \cdot - 1 t \cdot t - 8 t$

Simplify each term.

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

Move $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 \cdot - 1 (t \cdot t) - 8 t$

Multiply $t$ by $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 \cdot - 1 t^{2} - 8 t$

Multiply $- 4$ by $- 1$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} + 4 t^{2} - 8 t$

Simplify by adding terms.

Add $- 8 t^{4}$ and $12 t^{4}$ .

$4 t^{5} + 4 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} + 4 t^{2} - 8 t$

Add $- 24 t^{3}$ and $12 t^{3}$ .

$4 t^{5} + 4 t^{4} - 12 t^{3} - 24 t^{2} + 4 t^{2} - 8 t$

Add $- 24 t^{2}$ and $4 t^{2}$ .

$4 t^{5} + 4 t^{4} - 12 t^{3} - 20 t^{2} - 8 t$

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

$4 t^{5} \to 5$

$4 t^{4} \to 4$

$- 12 t^{3} \to 3$

$- 20 t^{2} \to 2$

$- 8 t \to 1$

The largest exponent is the degree of the polynomial.

$5$

Step 3

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

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

Simplify each term.

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

Move $t$ .

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

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

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

$- 4 (t^{1} t^{3}) (- t + 2) - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

$- 4 t^{1 + 3} (- t + 2) - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Add $1$ and $3$ .

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

Apply the distributive property.

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

Rewrite using the commutative property of multiplication.

$- 4 \cdot - 1 t^{4} t - 4 t^{4} \cdot 2 - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Multiply $2$ by $- 4$ .

$- 4 \cdot - 1 t^{4} t - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Simplify each term.

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

Move $t$ .

$- 4 \cdot - 1 (t \cdot t^{4}) - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

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

$- 4 \cdot - 1 (t^{1} t^{4}) - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

$- 4 \cdot - 1 t^{1 + 4} - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Add $1$ and $4$ .

$- 4 \cdot - 1 t^{5} - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Multiply $- 4$ by $- 1$ .

$4 t^{5} - 8 t^{4} - 12 t^{2} t (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

Move $t$ .

$4 t^{5} - 8 t^{4} - 12 (t \cdot t^{2}) (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

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

$4 t^{5} - 8 t^{4} - 12 (t^{1} t^{2}) (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

$4 t^{5} - 8 t^{4} - 12 t^{1 + 2} (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Add $1$ and $2$ .

$4 t^{5} - 8 t^{4} - 12 t^{3} (- t + 2) - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Apply the distributive property.

$4 t^{5} - 8 t^{4} - 12 t^{3} (- t) - 12 t^{3} \cdot 2 - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Rewrite using the commutative property of multiplication.

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 t^{3} t - 12 t^{3} \cdot 2 - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Multiply $2$ by $- 12$ .

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 t^{3} t - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Simplify each term.

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

Move $t$ .

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 (t \cdot t^{3}) - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

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

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 (t^{1} t^{3}) - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 t^{1 + 3} - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Add $1$ and $3$ .

$4 t^{5} - 8 t^{4} - 12 \cdot - 1 t^{4} - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

Multiply $- 12$ by $- 1$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 t \cdot t (- t + 2) - 4 t (- t + 2)$

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

Move $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 (t \cdot t) (- t + 2) - 4 t (- t + 2)$

Multiply $t$ by $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 t^{2} (- t + 2) - 4 t (- t + 2)$

Apply the distributive property.

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 t^{2} (- t) - 12 t^{2} \cdot 2 - 4 t (- t + 2)$

Rewrite using the commutative property of multiplication.

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 t^{2} t - 12 t^{2} \cdot 2 - 4 t (- t + 2)$

Multiply $2$ by $- 12$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 t^{2} t - 24 t^{2} - 4 t (- t + 2)$

Simplify each term.

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

Move $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 (t \cdot t^{2}) - 24 t^{2} - 4 t (- t + 2)$

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

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

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 (t^{1} t^{2}) - 24 t^{2} - 4 t (- t + 2)$

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

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 t^{1 + 2} - 24 t^{2} - 4 t (- t + 2)$

Add $1$ and $2$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} - 12 \cdot - 1 t^{3} - 24 t^{2} - 4 t (- t + 2)$

Multiply $- 12$ by $- 1$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 t (- t + 2)$

Apply the distributive property.

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 t (- t) - 4 t \cdot 2$

Rewrite using the commutative property of multiplication.

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 \cdot - 1 t \cdot t - 4 t \cdot 2$

Multiply $2$ by $- 4$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 \cdot - 1 t \cdot t - 8 t$

Simplify each term.

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

Move $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 \cdot - 1 (t \cdot t) - 8 t$

Multiply $t$ by $t$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} - 4 \cdot - 1 t^{2} - 8 t$

Multiply $- 4$ by $- 1$ .

$4 t^{5} - 8 t^{4} + 12 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} + 4 t^{2} - 8 t$

Simplify by adding terms.

Add $- 8 t^{4}$ and $12 t^{4}$ .

$4 t^{5} + 4 t^{4} - 24 t^{3} + 12 t^{3} - 24 t^{2} + 4 t^{2} - 8 t$

Add $- 24 t^{3}$ and $12 t^{3}$ .

$4 t^{5} + 4 t^{4} - 12 t^{3} - 24 t^{2} + 4 t^{2} - 8 t$

Add $- 24 t^{2}$ and $4 t^{2}$ .

$4 t^{5} + 4 t^{4} - 12 t^{3} - 20 t^{2} - 8 t$

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

$4 t^{5}$

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.

$4 t^{5}$

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

$4$

Step 5

List the results.

Polynomial Degree: $5$

Leading Term: $4 t^{5}$

Leading Coefficient: $4$

Find the Degree, Leading Term, and Leading Coefficient q(t)=-4t(2-t)(t+1)^3

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