Find the Degree, Leading Term, and Leading Coefficient h(t)=3(t-5)^3(t-3)^3(t-2)

$h (t) = 3 {(t - 5)}^{3} {(t - 3)}^{3} (t - 2)$

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

Use the Binomial Theorem.

$h (t) = 3 (t^{3} + 3 t^{2} \cdot - 5 + 3 t {(- 5)}^{2} + {(- 5)}^{3}) {(t - 3)}^{3} (t - 2)$

Simplify terms.

Simplify each term.

Multiply $- 5$ by $3$ .

$h (t) = 3 (t^{3} - 15 t^{2} + 3 t {(- 5)}^{2} + {(- 5)}^{3}) {(t - 3)}^{3} (t - 2)$

Raise $- 5$ to the power of $2$ .

$h (t) = 3 (t^{3} - 15 t^{2} + 3 t \cdot 25 + {(- 5)}^{3}) {(t - 3)}^{3} (t - 2)$

Multiply $25$ by $3$ .

$h (t) = 3 (t^{3} - 15 t^{2} + 75 t + {(- 5)}^{3}) {(t - 3)}^{3} (t - 2)$

Raise $- 5$ to the power of $3$ .

$h (t) = 3 (t^{3} - 15 t^{2} + 75 t - 125) {(t - 3)}^{3} (t - 2)$

Apply the distributive property.

$h (t) = (3 t^{3} + 3 (- 15 t^{2}) + 3 (75 t) + 3 \cdot - 125) {(t - 3)}^{3} (t - 2)$

Simplify.

Multiply $- 15$ by $3$ .

$h (t) = (3 t^{3} - 45 t^{2} + 3 (75 t) + 3 \cdot - 125) {(t - 3)}^{3} (t - 2)$

Multiply $75$ by $3$ .

$h (t) = (3 t^{3} - 45 t^{2} + 225 t + 3 \cdot - 125) {(t - 3)}^{3} (t - 2)$

Multiply $3$ by $- 125$ .

$h (t) = (3 t^{3} - 45 t^{2} + 225 t - 375) {(t - 3)}^{3} (t - 2)$

Use the Binomial Theorem.

$h (t) = (3 t^{3} - 45 t^{2} + 225 t - 375) (t^{3} + 3 t^{2} \cdot - 3 + 3 t {(- 3)}^{2} + {(- 3)}^{3}) (t - 2)$

Simplify each term.

Multiply $- 3$ by $3$ .

$h (t) = (3 t^{3} - 45 t^{2} + 225 t - 375) (t^{3} - 9 t^{2} + 3 t {(- 3)}^{2} + {(- 3)}^{3}) (t - 2)$

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

$h (t) = (3 t^{3} - 45 t^{2} + 225 t - 375) (t^{3} - 9 t^{2} + 3 t \cdot 9 + {(- 3)}^{3}) (t - 2)$

Multiply $9$ by $3$ .

$h (t) = (3 t^{3} - 45 t^{2} + 225 t - 375) (t^{3} - 9 t^{2} + 27 t + {(- 3)}^{3}) (t - 2)$

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

$h (t) = (3 t^{3} - 45 t^{2} + 225 t - 375) (t^{3} - 9 t^{2} + 27 t - 27) (t - 2)$

Expand $(3 t^{3} - 45 t^{2} + 225 t - 375) (t^{3} - 9 t^{2} + 27 t - 27)$ by multiplying each term in the first expression by each term in the second expression.

$h (t) = (3 t^{3} t^{3} + 3 t^{3} (- 9 t^{2}) + 3 t^{3} (27 t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Simplify terms.

Simplify each term.

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

Move $t^{3}$ .

$h (t) = (3 (t^{3} t^{3}) + 3 t^{3} (- 9 t^{2}) + 3 t^{3} (27 t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

$h (t) = (3 t^{3 + 3} + 3 t^{3} (- 9 t^{2}) + 3 t^{3} (27 t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Add $3$ and $3$ .

$h (t) = (3 t^{6} + 3 t^{3} (- 9 t^{2}) + 3 t^{3} (27 t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Rewrite using the commutative property of multiplication.

$h (t) = (3 t^{6} + 3 \cdot (- 9 t^{3} t^{2}) + 3 t^{3} (27 t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

Move $t^{2}$ .

$h (t) = (3 t^{6} + 3 \cdot (- 9 (t^{2} t^{3})) + 3 t^{3} (27 t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

$h (t) = (3 t^{6} + 3 \cdot (- 9 t^{2 + 3}) + 3 t^{3} (27 t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Add $2$ and $3$ .

$h (t) = (3 t^{6} + 3 \cdot (- 9 t^{5}) + 3 t^{3} (27 t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $3$ by $- 9$ .

$h (t) = (3 t^{6} - 27 t^{5} + 3 t^{3} (27 t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Rewrite using the commutative property of multiplication.

$h (t) = (3 t^{6} - 27 t^{5} + 3 \cdot (27 t^{3} t) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

Move $t$ .

$h (t) = (3 t^{6} - 27 t^{5} + 3 \cdot (27 (t \cdot t^{3})) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

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

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

$h (t) = (3 t^{6} - 27 t^{5} + 3 \cdot (27 t^{1 + 3}) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Add $1$ and $3$ .

$h (t) = (3 t^{6} - 27 t^{5} + 3 \cdot (27 t^{4}) + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $3$ by $27$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} + 3 t^{3} \cdot - 27 - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $- 27$ by $3$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{2} t^{3} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

Move $t^{3}$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 (t^{3} t^{2}) - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{3 + 2} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Add $3$ and $2$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} - 45 t^{2} (- 9 t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Rewrite using the commutative property of multiplication.

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} - 45 \cdot (- 9 t^{2} t^{2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

Move $t^{2}$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} - 45 \cdot (- 9 (t^{2} t^{2})) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} - 45 \cdot (- 9 t^{2 + 2}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Add $2$ and $2$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} - 45 \cdot (- 9 t^{4}) - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $- 45$ by $- 9$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 45 t^{2} (27 t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Rewrite using the commutative property of multiplication.

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 45 \cdot (27 t^{2} t) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

Move $t$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 45 \cdot (27 (t \cdot t^{2})) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

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

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

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 45 \cdot (27 t^{1 + 2}) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Add $1$ and $2$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 45 \cdot (27 t^{3}) - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $- 45$ by $27$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} - 45 t^{2} \cdot - 27 + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $- 27$ by $- 45$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t \cdot t^{3} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

Move $t^{3}$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 (t^{3} t) + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

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

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

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{3 + 1} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Add $3$ and $1$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} + 225 t (- 9 t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Rewrite using the commutative property of multiplication.

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} + 225 \cdot (- 9 t \cdot t^{2}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

Move $t^{2}$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} + 225 \cdot (- 9 (t^{2} t)) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

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

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

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} + 225 \cdot (- 9 t^{2 + 1}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Add $2$ and $1$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} + 225 \cdot (- 9 t^{3}) + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $225$ by $- 9$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 225 t (27 t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Rewrite using the commutative property of multiplication.

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 225 \cdot (27 t \cdot t) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

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

Move $t$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 225 \cdot (27 (t \cdot t)) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $t$ by $t$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 225 \cdot (27 t^{2}) + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $225$ by $27$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 6075 t^{2} + 225 t \cdot - 27 - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $- 27$ by $225$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 6075 t^{2} - 6075 t - 375 t^{3} - 375 (- 9 t^{2}) - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $- 9$ by $- 375$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 6075 t^{2} - 6075 t - 375 t^{3} + 3375 t^{2} - 375 (27 t) - 375 \cdot - 27) (t - 2)$

Multiply $27$ by $- 375$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 6075 t^{2} - 6075 t - 375 t^{3} + 3375 t^{2} - 10125 t - 375 \cdot - 27) (t - 2)$

Multiply $- 375$ by $- 27$ .

$h (t) = (3 t^{6} - 27 t^{5} + 81 t^{4} - 81 t^{3} - 45 t^{5} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 6075 t^{2} - 6075 t - 375 t^{3} + 3375 t^{2} - 10125 t + 10125) (t - 2)$

Simplify by adding terms.

Subtract $45 t^{5}$ from $- 27 t^{5}$ .

$h (t) = (3 t^{6} - 72 t^{5} + 81 t^{4} - 81 t^{3} + 405 t^{4} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 6075 t^{2} - 6075 t - 375 t^{3} + 3375 t^{2} - 10125 t + 10125) (t - 2)$

Add $81 t^{4}$ and $405 t^{4}$ .

$h (t) = (3 t^{6} - 72 t^{5} + 486 t^{4} - 81 t^{3} - 1215 t^{3} + 1215 t^{2} + 225 t^{4} - 2025 t^{3} + 6075 t^{2} - 6075 t - 375 t^{3} + 3375 t^{2} - 10125 t + 10125) (t - 2)$

Add $486 t^{4}$ and $225 t^{4}$ .

$h (t) = (3 t^{6} - 72 t^{5} + 711 t^{4} - 81 t^{3} - 1215 t^{3} + 1215 t^{2} - 2025 t^{3} + 6075 t^{2} - 6075 t - 375 t^{3} + 3375 t^{2} - 10125 t + 10125) (t - 2)$

Subtract $1215 t^{3}$ from $- 81 t^{3}$ .

$h (t) = (3 t^{6} - 72 t^{5} + 711 t^{4} - 1296 t^{3} + 1215 t^{2} - 2025 t^{3} + 6075 t^{2} - 6075 t - 375 t^{3} + 3375 t^{2} - 10125 t + 10125) (t - 2)$

Subtract $2025 t^{3}$ from $- 1296 t^{3}$ .

$h (t) = (3 t^{6} - 72 t^{5} + 711 t^{4} - 3321 t^{3} + 1215 t^{2} + 6075 t^{2} - 6075 t - 375 t^{3} + 3375 t^{2} - 10125 t + 10125) (t - 2)$

Subtract $375 t^{3}$ from $- 3321 t^{3}$ .

$h (t) = (3 t^{6} - 72 t^{5} + 711 t^{4} - 3696 t^{3} + 1215 t^{2} + 6075 t^{2} - 6075 t + 3375 t^{2} - 10125 t + 10125) (t - 2)$

Add $1215 t^{2}$ and $6075 t^{2}$ .

$h (t) = (3 t^{6} - 72 t^{5} + 711 t^{4} - 3696 t^{3} + 7290 t^{2} - 6075 t + 3375 t^{2} - 10125 t + 10125) (t - 2)$

Add $7290 t^{2}$ and $3375 t^{2}$ .

$h (t) = (3 t^{6} - 72 t^{5} + 711 t^{4} - 3696 t^{3} + 10665 t^{2} - 6075 t - 10125 t + 10125) (t - 2)$

Subtract $10125 t$ from $- 6075 t$ .

$h (t) = (3 t^{6} - 72 t^{5} + 711 t^{4} - 3696 t^{3} + 10665 t^{2} - 16200 t + 10125) (t - 2)$

Expand $(3 t^{6} - 72 t^{5} + 711 t^{4} - 3696 t^{3} + 10665 t^{2} - 16200 t + 10125) (t - 2)$ by multiplying each term in the first expression by each term in the second expression.

$h (t) = 3 t^{6} t + 3 t^{6} \cdot - 2 - 72 t^{5} t - 72 t^{5} \cdot - 2 + 711 t^{4} t + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Simplify terms.

Simplify each term.

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

Move $t$ .

$h (t) = 3 (t \cdot t^{6}) + 3 t^{6} \cdot - 2 - 72 t^{5} t - 72 t^{5} \cdot - 2 + 711 t^{4} t + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

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

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

$h (t) = 3 t^{1 + 6} + 3 t^{6} \cdot - 2 - 72 t^{5} t - 72 t^{5} \cdot - 2 + 711 t^{4} t + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Add $1$ and $6$ .

$h (t) = 3 t^{7} + 3 t^{6} \cdot - 2 - 72 t^{5} t - 72 t^{5} \cdot - 2 + 711 t^{4} t + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Multiply $- 2$ by $3$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{5} t - 72 t^{5} \cdot - 2 + 711 t^{4} t + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

Move $t$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 (t \cdot t^{5}) - 72 t^{5} \cdot - 2 + 711 t^{4} t + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

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

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

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{1 + 5} - 72 t^{5} \cdot - 2 + 711 t^{4} t + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Add $1$ and $5$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} - 72 t^{5} \cdot - 2 + 711 t^{4} t + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Multiply $- 2$ by $- 72$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{4} t + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

Move $t$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 (t \cdot t^{4}) + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

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

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

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{1 + 4} + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Add $1$ and $4$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} + 711 t^{4} \cdot - 2 - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Multiply $- 2$ by $711$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{3} t - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

Move $t$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 (t \cdot t^{3}) - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

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

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

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{1 + 3} - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Add $1$ and $3$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} - 3696 t^{3} \cdot - 2 + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Multiply $- 2$ by $- 3696$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{2} t + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

Move $t$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 (t \cdot t^{2}) + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

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

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

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{1 + 2} + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Add $1$ and $2$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{3} + 10665 t^{2} \cdot - 2 - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Multiply $- 2$ by $10665$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{3} - 21330 t^{2} - 16200 t \cdot t - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

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

Move $t$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{3} - 21330 t^{2} - 16200 (t \cdot t) - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Multiply $t$ by $t$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{3} - 21330 t^{2} - 16200 t^{2} - 16200 t \cdot - 2 + 10125 t + 10125 \cdot - 2$

Multiply $- 2$ by $- 16200$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{3} - 21330 t^{2} - 16200 t^{2} + 32400 t + 10125 t + 10125 \cdot - 2$

Multiply $10125$ by $- 2$ .

$h (t) = 3 t^{7} - 6 t^{6} - 72 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{3} - 21330 t^{2} - 16200 t^{2} + 32400 t + 10125 t - 20250$

Simplify by adding terms.

Subtract $72 t^{6}$ from $- 6 t^{6}$ .

$h (t) = 3 t^{7} - 78 t^{6} + 144 t^{5} + 711 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{3} - 21330 t^{2} - 16200 t^{2} + 32400 t + 10125 t - 20250$

Add $144 t^{5}$ and $711 t^{5}$ .

$h (t) = 3 t^{7} - 78 t^{6} + 855 t^{5} - 1422 t^{4} - 3696 t^{4} + 7392 t^{3} + 10665 t^{3} - 21330 t^{2} - 16200 t^{2} + 32400 t + 10125 t - 20250$

Subtract $3696 t^{4}$ from $- 1422 t^{4}$ .

$h (t) = 3 t^{7} - 78 t^{6} + 855 t^{5} - 5118 t^{4} + 7392 t^{3} + 10665 t^{3} - 21330 t^{2} - 16200 t^{2} + 32400 t + 10125 t - 20250$

Add $7392 t^{3}$ and $10665 t^{3}$ .

$h (t) = 3 t^{7} - 78 t^{6} + 855 t^{5} - 5118 t^{4} + 18057 t^{3} - 21330 t^{2} - 16200 t^{2} + 32400 t + 10125 t - 20250$

Subtract $16200 t^{2}$ from $- 21330 t^{2}$ .

$h (t) = 3 t^{7} - 78 t^{6} + 855 t^{5} - 5118 t^{4} + 18057 t^{3} - 37530 t^{2} + 32400 t + 10125 t - 20250$

Add $32400 t$ and $10125 t$ .

$h (t) = 3 t^{7} - 78 t^{6} + 855 t^{5} - 5118 t^{4} + 18057 t^{3} - 37530 t^{2} + 42525 t - 20250$

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.

$3 t^{7} \to 7$

$- 78 t^{6} \to 6$

$855 t^{5} \to 5$

$- 5118 t^{4} \to 4$

$18057 t^{3} \to 3$

$- 37530 t^{2} \to 2$

$42525 t \to 1$

$- 20250 \to 0$

The largest exponent is the degree of the polynomial.

$7$

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

$3 t^{7}$

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.

$3 t^{7}$

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

$3$

List the results.

Polynomial Degree: $7$

Leading Term: $3 t^{7}$

Leading Coefficient: $3$

Find the Degree, Leading Term, and Leading Coefficient h(t)=3(t-5)^3(t-3)^3(t-2)

Related Questions

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