Algebra Examples

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

Step 1

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

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $- 5$ by $3$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Multiply $25$ by $3$ .

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

Step 1.2.1.4

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

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

Step 1.2.2

Apply the distributive property.

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $- 15$ by $3$ .

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

Step 1.3.2

Multiply $75$ by $3$ .

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

Step 1.3.3

Multiply $3$ by $- 125$ .

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

Step 1.4

Use the Binomial Theorem.

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

Step 1.5

Simplify each term.

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Step 1.5.1

Multiply $- 3$ by $3$ .

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

Step 1.5.2

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

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

Step 1.5.3

Multiply $9$ by $3$ .

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

Step 1.5.4

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

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

Step 1.6

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.

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

Step 1.7

Simplify terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

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

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Step 1.7.1.1.1

Move $t^{3}$ .

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

Step 1.7.1.1.2

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

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

Step 1.7.1.1.3

Add $3$ and $3$ .

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

Step 1.7.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.3

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

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Step 1.7.1.3.1

Move $t^{2}$ .

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

Step 1.7.1.3.2

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

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

Step 1.7.1.3.3

Add $2$ and $3$ .

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

Step 1.7.1.4

Multiply $3$ by $- 9$ .

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

Step 1.7.1.5

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.6

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

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Step 1.7.1.6.1

Move $t$ .

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

Step 1.7.1.6.2

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

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Step 1.7.1.6.2.1

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

Step 1.7.1.6.2.2

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

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

Step 1.7.1.6.3

Add $1$ and $3$ .

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

Step 1.7.1.7

Multiply $3$ by $27$ .

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

Step 1.7.1.8

Multiply $- 27$ by $3$ .

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

Step 1.7.1.9

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

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Step 1.7.1.9.1

Move $t^{3}$ .

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

Step 1.7.1.9.2

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

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

Step 1.7.1.9.3

Add $3$ and $2$ .

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

Step 1.7.1.10

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.11

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

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Step 1.7.1.11.1

Move $t^{2}$ .

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

Step 1.7.1.11.2

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

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

Step 1.7.1.11.3

Add $2$ and $2$ .

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

Step 1.7.1.12

Multiply $- 45$ by $- 9$ .

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

Step 1.7.1.13

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.14

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

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Step 1.7.1.14.1

Move $t$ .

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

Step 1.7.1.14.2

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

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Step 1.7.1.14.2.1

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

Step 1.7.1.14.2.2

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

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

Step 1.7.1.14.3

Add $1$ and $2$ .

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

Step 1.7.1.15

Multiply $- 45$ by $27$ .

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

Step 1.7.1.16

Multiply $- 27$ by $- 45$ .

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

Step 1.7.1.17

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

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Step 1.7.1.17.1

Move $t^{3}$ .

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

Step 1.7.1.17.2

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

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Step 1.7.1.17.2.1

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

Step 1.7.1.17.2.2

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

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

Step 1.7.1.17.3

Add $3$ and $1$ .

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

Step 1.7.1.18

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.19

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

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Step 1.7.1.19.1

Move $t^{2}$ .

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

Step 1.7.1.19.2

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

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Step 1.7.1.19.2.1

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

Step 1.7.1.19.2.2

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

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

Step 1.7.1.19.3

Add $2$ and $1$ .

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

Step 1.7.1.20

Multiply $225$ by $- 9$ .

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

Step 1.7.1.21

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.22

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

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Step 1.7.1.22.1

Move $t$ .

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

Step 1.7.1.22.2

Multiply $t$ by $t$ .

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

Step 1.7.1.23

Multiply $225$ by $27$ .

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

Step 1.7.1.24

Multiply $- 27$ by $225$ .

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

Step 1.7.1.25

Multiply $- 9$ by $- 375$ .

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

Step 1.7.1.26

Multiply $27$ by $- 375$ .

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

Step 1.7.1.27

Multiply $- 375$ by $- 27$ .

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

Step 1.7.2

Simplify by adding terms.

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Step 1.7.2.1

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

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

Step 1.7.2.2

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

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

Step 1.7.2.3

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

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

Step 1.7.2.4

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

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

Step 1.7.2.5

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

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

Step 1.7.2.6

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

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

Step 1.7.2.7

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

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

Step 1.7.2.8

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

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

Step 1.7.2.9

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

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

Step 1.8

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.

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

Step 1.9

Simplify terms.

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

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

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Step 1.9.1.1.1

Move $t$ .

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

Step 1.9.1.1.2

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

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Step 1.9.1.1.2.1

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

Step 1.9.1.1.2.2

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

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

Step 1.9.1.1.3

Add $1$ and $6$ .

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

Step 1.9.1.2

Multiply $- 2$ by $3$ .

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

Step 1.9.1.3

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

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Step 1.9.1.3.1

Move $t$ .

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

Step 1.9.1.3.2

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

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Step 1.9.1.3.2.1

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

Step 1.9.1.3.2.2

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

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

Step 1.9.1.3.3

Add $1$ and $5$ .

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

Step 1.9.1.4

Multiply $- 2$ by $- 72$ .

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

Step 1.9.1.5

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

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Step 1.9.1.5.1

Move $t$ .

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

Step 1.9.1.5.2

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

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Step 1.9.1.5.2.1

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

Step 1.9.1.5.2.2

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

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

Step 1.9.1.5.3

Add $1$ and $4$ .

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

Step 1.9.1.6

Multiply $- 2$ by $711$ .

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

Step 1.9.1.7

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

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Step 1.9.1.7.1

Move $t$ .

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

Step 1.9.1.7.2

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

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Step 1.9.1.7.2.1

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

Step 1.9.1.7.2.2

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

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

Step 1.9.1.7.3

Add $1$ and $3$ .

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

Step 1.9.1.8

Multiply $- 2$ by $- 3696$ .

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

Step 1.9.1.9

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

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Step 1.9.1.9.1

Move $t$ .

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

Step 1.9.1.9.2

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

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Step 1.9.1.9.2.1

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

Step 1.9.1.9.2.2

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

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

Step 1.9.1.9.3

Add $1$ and $2$ .

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

Step 1.9.1.10

Multiply $- 2$ by $10665$ .

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

Step 1.9.1.11

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

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Step 1.9.1.11.1

Move $t$ .

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

Step 1.9.1.11.2

Multiply $t$ by $t$ .

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

Step 1.9.1.12

Multiply $- 2$ by $- 16200$ .

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

Step 1.9.1.13

Multiply $10125$ by $- 2$ .

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

Step 1.9.2

Simplify by adding terms.

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Step 1.9.2.1

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

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

Step 1.9.2.2

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

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

Step 1.9.2.3

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

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

Step 1.9.2.4

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

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

Step 1.9.2.5

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

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

Step 1.9.2.6

Add $32400 t$ and $10125 t$ .

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

Step 2

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

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Step 2.1

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

$3 t^{7} \to 7$

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

$855 t^{5} \to 5$

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

$18057 t^{3} \to 3$

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

$42525 t \to 1$

$- 20250 \to 0$

Step 2.2

The largest exponent is the degree of the polynomial.

$7$

Step 3

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

$3 t^{7}$

Step 4

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

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Step 4.1

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

$3 t^{7}$

Step 4.2

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

$3$

Step 5

List the results.

Polynomial Degree: $7$

Leading Term: $3 t^{7}$

Leading Coefficient: $3$