Algebra Examples

$f (t) = \frac{1}{10} t^{2} {(t - 4)}^{3} {(t + 5)}^{2}$

Step 1

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

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Step 1.1

Combine $\frac{1}{10}$ and $t^{2}$ .

$f (t) = \frac{t^{2}}{10} \cdot ({(t - 4)}^{3} {(t + 5)}^{2})$

Step 1.2

Use the Binomial Theorem.

$f (t) = \frac{t^{2}}{10} \cdot ((t^{3} + 3 t^{2} \cdot - 4 + 3 t {(- 4)}^{2} + {(- 4)}^{3}) {(t + 5)}^{2})$

Step 1.3

Simplify terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $- 4$ by $3$ .

$f (t) = \frac{t^{2}}{10} \cdot ((t^{3} - 12 t^{2} + 3 t {(- 4)}^{2} + {(- 4)}^{3}) {(t + 5)}^{2})$

Step 1.3.1.2

Raise $- 4$ to the power of $2$ .

$f (t) = \frac{t^{2}}{10} \cdot ((t^{3} - 12 t^{2} + 3 t \cdot 16 + {(- 4)}^{3}) {(t + 5)}^{2})$

Step 1.3.1.3

Multiply $16$ by $3$ .

$f (t) = \frac{t^{2}}{10} \cdot ((t^{3} - 12 t^{2} + 48 t + {(- 4)}^{3}) {(t + 5)}^{2})$

Step 1.3.1.4

Raise $- 4$ to the power of $3$ .

$f (t) = \frac{t^{2}}{10} \cdot ((t^{3} - 12 t^{2} + 48 t - 64) {(t + 5)}^{2})$

Step 1.3.2

Apply the distributive property.

$f (t) = (\frac{t^{2}}{10} \cdot t^{3} + \frac{t^{2}}{10} \cdot (- 12 t^{2}) + \frac{t^{2}}{10} \cdot (48 t) + \frac{t^{2}}{10} \cdot - 64) {(t + 5)}^{2}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $\frac{t^{2}}{10} t^{3}$ .

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Step 1.4.1.1

Combine $\frac{t^{2}}{10}$ and $t^{3}$ .

$f (t) = (\frac{t^{2} t^{3}}{10} + \frac{t^{2}}{10} \cdot (- 12 t^{2}) + \frac{t^{2}}{10} \cdot (48 t) + \frac{t^{2}}{10} \cdot - 64) {(t + 5)}^{2}$

Step 1.4.1.2

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

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Step 1.4.1.2.1

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

$f (t) = (\frac{t^{2 + 3}}{10} + \frac{t^{2}}{10} \cdot (- 12 t^{2}) + \frac{t^{2}}{10} \cdot (48 t) + \frac{t^{2}}{10} \cdot - 64) {(t + 5)}^{2}$

Step 1.4.1.2.2

Add $2$ and $3$ .

$f (t) = (\frac{t^{5}}{10} + \frac{t^{2}}{10} \cdot (- 12 t^{2}) + \frac{t^{2}}{10} \cdot (48 t) + \frac{t^{2}}{10} \cdot - 64) {(t + 5)}^{2}$

Step 1.4.2

Rewrite using the commutative property of multiplication.

$f (t) = (\frac{t^{5}}{10} - 12 \frac{t^{2}}{10} \cdot t^{2} + \frac{t^{2}}{10} \cdot (48 t) + \frac{t^{2}}{10} \cdot - 64) {(t + 5)}^{2}$

Step 1.4.3

Rewrite using the commutative property of multiplication.

$f (t) = (\frac{t^{5}}{10} - 12 \frac{t^{2}}{10} \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2}}{10} \cdot - 64) {(t + 5)}^{2}$

Step 1.4.4

Cancel the common factor of $2$ .

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Step 1.4.4.1

Factor $2$ out of $10$ .

$f (t) = (\frac{t^{5}}{10} - 12 \frac{t^{2}}{10} \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2}}{2 (5)} \cdot - 64) {(t + 5)}^{2}$

Step 1.4.4.2

Factor $2$ out of $- 64$ .

$f (t) = (\frac{t^{5}}{10} - 12 \frac{t^{2}}{10} \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2}}{2 \cdot 5} \cdot (2 \cdot - 32)) {(t + 5)}^{2}$

Step 1.4.4.3

Cancel the common factor.

$f (t) = (\frac{t^{5}}{10} - 12 \frac{t^{2}}{10} \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2}}{2 \cdot 5} \cdot (2 \cdot - 32)) {(t + 5)}^{2}$

Step 1.4.4.4

Rewrite the expression.

$f (t) = (\frac{t^{5}}{10} - 12 \frac{t^{2}}{10} \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2}}{5} \cdot - 32) {(t + 5)}^{2}$

Step 1.4.5

Combine $\frac{t^{2}}{5}$ and $- 32$ .

$f (t) = (\frac{t^{5}}{10} - 12 \frac{t^{2}}{10} \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Cancel the common factor of $2$ .

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Step 1.5.1.1.1

Factor $2$ out of $- 12$ .

$f (t) = (\frac{t^{5}}{10} + 2 (- 6) (\frac{t^{2}}{10}) \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.1.2

Factor $2$ out of $10$ .

$f (t) = (\frac{t^{5}}{10} + 2 \cdot (- 6 \frac{t^{2}}{2 \cdot 5}) \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.1.3

Cancel the common factor.

$f (t) = (\frac{t^{5}}{10} + 2 \cdot (- 6 \frac{t^{2}}{2 \cdot 5}) \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.1.4

Rewrite the expression.

$f (t) = (\frac{t^{5}}{10} - 6 \frac{t^{2}}{5} \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.2

Combine $- 6$ and $\frac{t^{2}}{5}$ .

$f (t) = (\frac{t^{5}}{10} + \frac{- 6 t^{2}}{5} \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.3

Move the negative in front of the fraction.

$f (t) = (\frac{t^{5}}{10} - \frac{(6) t^{2}}{5} \cdot t^{2} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.4

Multiply $- \frac{6 t^{2}}{5} t^{2}$ .

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Step 1.5.1.4.1

Combine $t^{2}$ and $\frac{6 t^{2}}{5}$ .

$f (t) = (\frac{t^{5}}{10} - \frac{t^{2} (6 t^{2})}{5} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.4.2

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

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Step 1.5.1.4.2.1

Move $t^{2}$ .

$f (t) = (\frac{t^{5}}{10} - \frac{t^{2} t^{2} \cdot 6}{5} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.4.2.2

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

$f (t) = (\frac{t^{5}}{10} - \frac{t^{2 + 2} \cdot 6}{5} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.4.2.3

Add $2$ and $2$ .

$f (t) = (\frac{t^{5}}{10} - \frac{t^{4} \cdot 6}{5} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.5

Move $6$ to the left of $t^{4}$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 \cdot t^{4}}{5} + 48 (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.6

Cancel the common factor of $2$ .

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Step 1.5.1.6.1

Factor $2$ out of $48$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + 2 (24) (\frac{t^{2}}{10}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.6.2

Factor $2$ out of $10$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + 2 \cdot (24 (\frac{t^{2}}{2 \cdot 5})) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.6.3

Cancel the common factor.

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + 2 \cdot (24 (\frac{t^{2}}{2 \cdot 5})) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.6.4

Rewrite the expression.

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + 24 (\frac{t^{2}}{5}) \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.7

Combine $24$ and $\frac{t^{2}}{5}$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{2}}{5} \cdot t + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.8

Multiply $\frac{24 t^{2}}{5} t$ .

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Step 1.5.1.8.1

Combine $\frac{24 t^{2}}{5}$ and $t$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{2} t}{5} + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.8.2

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

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 (t \cdot t^{2})}{5} + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.8.3

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

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{1 + 2}}{5} + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.8.4

Add $1$ and $2$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} + \frac{t^{2} \cdot - 32}{5}) {(t + 5)}^{2}$

Step 1.5.1.9

Move $- 32$ to the left of $t^{2}$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} + \frac{- 32 \cdot t^{2}}{5}) {(t + 5)}^{2}$

Step 1.5.1.10

Move the negative in front of the fraction.

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) {(t + 5)}^{2}$

Step 1.5.2

Rewrite ${(t + 5)}^{2}$ as $(t + 5) (t + 5)$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) ((t + 5) (t + 5))$

Step 1.6

Expand $(t + 5) (t + 5)$ using the FOIL Method.

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Step 1.6.1

Apply the distributive property.

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) (t (t + 5) + 5 (t + 5))$

Step 1.6.2

Apply the distributive property.

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) (t \cdot t + t \cdot 5 + 5 (t + 5))$

Step 1.6.3

Apply the distributive property.

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) (t \cdot t + t \cdot 5 + 5 t + 5 \cdot 5)$

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $t$ by $t$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) (t^{2} + t \cdot 5 + 5 t + 5 \cdot 5)$

Step 1.7.1.2

Move $5$ to the left of $t$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) (t^{2} + 5 \cdot t + 5 t + 5 \cdot 5)$

Step 1.7.1.3

Multiply $5$ by $5$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) (t^{2} + 5 t + 5 t + 25)$

Step 1.7.2

Add $5 t$ and $5 t$ .

$f (t) = (\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) (t^{2} + 10 t + 25)$

Step 1.8

Expand $(\frac{t^{5}}{10} - \frac{6 t^{4}}{5} + \frac{24 t^{3}}{5} - \frac{32 t^{2}}{5}) (t^{2} + 10 t + 25)$ by multiplying each term in the first expression by each term in the second expression.

$f (t) = \frac{t^{5}}{10} \cdot t^{2} + \frac{t^{5}}{10} \cdot (10 t) + \frac{t^{5}}{10} \cdot 25 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9

Simplify each term.

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Step 1.9.1

Multiply $\frac{t^{5}}{10} t^{2}$ .

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Step 1.9.1.1

Combine $\frac{t^{5}}{10}$ and $t^{2}$ .

$f (t) = \frac{t^{5} t^{2}}{10} + \frac{t^{5}}{10} \cdot (10 t) + \frac{t^{5}}{10} \cdot 25 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.1.2

Multiply $t^{5}$ by $t^{2}$ by adding the exponents.

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Step 1.9.1.2.1

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

$f (t) = \frac{t^{5 + 2}}{10} + \frac{t^{5}}{10} \cdot (10 t) + \frac{t^{5}}{10} \cdot 25 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.1.2.2

Add $5$ and $2$ .

$f (t) = \frac{t^{7}}{10} + \frac{t^{5}}{10} \cdot (10 t) + \frac{t^{5}}{10} \cdot 25 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.2

Rewrite using the commutative property of multiplication.

$f (t) = \frac{t^{7}}{10} + 10 (\frac{t^{5}}{10}) \cdot t + \frac{t^{5}}{10} \cdot 25 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.3

Cancel the common factor of $10$ .

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Step 1.9.3.1

Cancel the common factor.

Step 1.9.3.2

Rewrite the expression.

$f (t) = \frac{t^{7}}{10} + t^{5} t + \frac{t^{5}}{10} \cdot 25 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.4

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

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Step 1.9.4.1

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

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Step 1.9.4.1.1

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

Step 1.9.4.1.2

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

$f (t) = \frac{t^{7}}{10} + t^{5 + 1} + \frac{t^{5}}{10} \cdot 25 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.4.2

Add $5$ and $1$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{t^{5}}{10} \cdot 25 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.5

Cancel the common factor of $5$ .

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Step 1.9.5.1

Factor $5$ out of $10$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{t^{5}}{5 (2)} \cdot 25 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.5.2

Factor $5$ out of $25$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{t^{5}}{5 \cdot 2} \cdot (5 \cdot 5) - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.5.3

Cancel the common factor.

Step 1.9.5.4

Rewrite the expression.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{t^{5}}{2} \cdot 5 - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.6

Combine $\frac{t^{5}}{2}$ and $5$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{t^{5} \cdot 5}{2} - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.7

Move $5$ to the left of $t^{5}$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 \cdot t^{5}}{2} - \frac{6 t^{4}}{5} \cdot t^{2} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.8

Multiply $- \frac{6 t^{4}}{5} t^{2}$ .

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Step 1.9.8.1

Combine $t^{2}$ and $\frac{6 t^{4}}{5}$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{t^{2} (6 t^{4})}{5} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.8.2

Multiply $t^{2}$ by $t^{4}$ by adding the exponents.

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Step 1.9.8.2.1

Move $t^{4}$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{t^{4} t^{2} \cdot 6}{5} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.8.2.2

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

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{t^{4 + 2} \cdot 6}{5} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.8.2.3

Add $4$ and $2$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{t^{6} \cdot 6}{5} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.9

Move $6$ to the left of $t^{6}$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 \cdot t^{6}}{5} - \frac{6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.10

Cancel the common factor of $5$ .

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Step 1.9.10.1

Move the leading negative in $- \frac{6 t^{4}}{5}$ into the numerator.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} + \frac{- 6 t^{4}}{5} \cdot (10 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.10.2

Factor $5$ out of $10 t$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} + \frac{- 6 t^{4}}{5} \cdot (5 (2 t)) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.10.3

Cancel the common factor.

Step 1.9.10.4

Rewrite the expression.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 6 t^{4} (2 t) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.11

Multiply $2$ by $- 6$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{4} t - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.12

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

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 (t \cdot t^{4}) - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.13

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

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{1 + 4} - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.14

Add $1$ and $4$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - \frac{6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.15

Cancel the common factor of $5$ .

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Step 1.9.15.1

Move the leading negative in $- \frac{6 t^{4}}{5}$ into the numerator.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} + \frac{- 6 t^{4}}{5} \cdot 25 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.15.2

Factor $5$ out of $25$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} + \frac{- 6 t^{4}}{5} \cdot (5 (5)) + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.15.3

Cancel the common factor.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} + \frac{- 6 t^{4}}{5} \cdot (5 \cdot 5) + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.15.4

Rewrite the expression.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 6 t^{4} \cdot 5 + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.16

Multiply $5$ by $- 6$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{3}}{5} \cdot t^{2} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.17

Multiply $\frac{24 t^{3}}{5} t^{2}$ .

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Step 1.9.17.1

Combine $\frac{24 t^{3}}{5}$ and $t^{2}$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{3} t^{2}}{5} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.17.2

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

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Step 1.9.17.2.1

Move $t^{2}$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 (t^{2} t^{3})}{5} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.17.2.2

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

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{2 + 3}}{5} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.17.2.3

Add $2$ and $3$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + \frac{24 t^{3}}{5} \cdot (10 t) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.18

Rewrite using the commutative property of multiplication.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 10 (\frac{24 t^{3}}{5}) \cdot t + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.19

Cancel the common factor of $5$ .

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Step 1.9.19.1

Factor $5$ out of $10$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 5 (2) (\frac{24 t^{3}}{5}) \cdot t + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.19.2

Cancel the common factor.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 5 \cdot (2 (\frac{24 t^{3}}{5})) \cdot t + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.19.3

Rewrite the expression.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 2 (24 t^{3}) t + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.20

Multiply $24$ by $2$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{3} t + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.21

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

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Step 1.9.21.1

Move $t$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 (t \cdot t^{3}) + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.21.2

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

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Step 1.9.21.2.1

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

Step 1.9.21.2.2

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

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{1 + 3} + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.21.3

Add $1$ and $3$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + \frac{24 t^{3}}{5} \cdot 25 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.22

Cancel the common factor of $5$ .

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Step 1.9.22.1

Factor $5$ out of $25$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + \frac{24 t^{3}}{5} \cdot (5 (5)) - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.22.2

Cancel the common factor.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + \frac{24 t^{3}}{5} \cdot (5 \cdot 5) - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.22.3

Rewrite the expression.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 24 t^{3} \cdot 5 - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.23

Multiply $5$ by $24$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{2}}{5} \cdot t^{2} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.24

Multiply $- \frac{32 t^{2}}{5} t^{2}$ .

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Step 1.9.24.1

Combine $t^{2}$ and $\frac{32 t^{2}}{5}$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{t^{2} (32 t^{2})}{5} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.24.2

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

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Step 1.9.24.2.1

Move $t^{2}$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{t^{2} t^{2} \cdot 32}{5} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.24.2.2

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

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{t^{2 + 2} \cdot 32}{5} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.24.2.3

Add $2$ and $2$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{t^{4} \cdot 32}{5} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.25

Move $32$ to the left of $t^{4}$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 \cdot t^{4}}{5} - \frac{32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.26

Cancel the common factor of $5$ .

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Step 1.9.26.1

Move the leading negative in $- \frac{32 t^{2}}{5}$ into the numerator.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} + \frac{- 32 t^{2}}{5} \cdot (10 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.26.2

Factor $5$ out of $10 t$ .

Step 1.9.26.3

Cancel the common factor.

Step 1.9.26.4

Rewrite the expression.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 32 t^{2} (2 t) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.27

Multiply $2$ by $- 32$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 t^{2} t - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.28

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

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 (t \cdot t^{2}) - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.29

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

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 t^{1 + 2} - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.30

Add $1$ and $2$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 t^{3} - \frac{32 t^{2}}{5} \cdot 25$

Step 1.9.31

Cancel the common factor of $5$ .

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Step 1.9.31.1

Move the leading negative in $- \frac{32 t^{2}}{5}$ into the numerator.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 t^{3} + \frac{- 32 t^{2}}{5} \cdot 25$

Step 1.9.31.2

Factor $5$ out of $25$ .

Step 1.9.31.3

Cancel the common factor.

Step 1.9.31.4

Rewrite the expression.

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 t^{3} - 32 t^{2} \cdot 5$

Step 1.9.32

Multiply $5$ by $- 32$ .

$f (t) = \frac{t^{7}}{10} + t^{6} + \frac{5 t^{5}}{2} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 t^{3} - 160 t^{2}$

Step 1.10

To write $t^{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (t) = \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + t^{6} \cdot \frac{5}{5} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 t^{3} - 160 t^{2}$

Step 1.11

Simplify terms.

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Step 1.11.1

Combine $t^{6}$ and $\frac{5}{5}$ .

$f (t) = \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{t^{6} \cdot 5}{5} - \frac{6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 t^{3} - 160 t^{2}$

Step 1.11.2

Combine the numerators over the common denominator.

$f (t) = \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{t^{6} \cdot 5 - 6 t^{6}}{5} - 12 t^{5} - 30 t^{4} + \frac{24 t^{5}}{5} + 48 t^{4} + 120 t^{3} - \frac{32 t^{4}}{5} - 64 t^{3} - 160 t^{2}$

Step 1.11.3

Combine the numerators over the common denominator.

$f (t) = - 12 t^{5} - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{t^{6} \cdot 5 - 6 t^{6} + 24 t^{5} - 32 t^{4}}{5}$

Step 1.12

Move $5$ to the left of $t^{6}$ .

$f (t) = - 12 t^{5} - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{5 t^{6} - 6 t^{6} + 24 t^{5} - 32 t^{4}}{5}$

Step 1.13

Simplify terms.

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Step 1.13.1

Subtract $6 t^{6}$ from $5 t^{6}$ .

$f (t) = - 12 t^{5} - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{- t^{6} + 24 t^{5} - 32 t^{4}}{5}$

Step 1.13.2

Factor $t^{4}$ out of $- t^{6} + 24 t^{5} - 32 t^{4}$ .

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Step 1.13.2.1

Factor $t^{4}$ out of $- t^{6}$ .

$f (t) = - 12 t^{5} - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{t^{4} (- t^{2}) + 24 t^{5} - 32 t^{4}}{5}$

Step 1.13.2.2

Factor $t^{4}$ out of $24 t^{5}$ .

$f (t) = - 12 t^{5} - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{t^{4} (- t^{2}) + t^{4} (24 t) - 32 t^{4}}{5}$

Step 1.13.2.3

Factor $t^{4}$ out of $- 32 t^{4}$ .

$f (t) = - 12 t^{5} - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{t^{4} (- t^{2}) + t^{4} (24 t) + t^{4} \cdot - 32}{5}$

Step 1.13.2.4

Factor $t^{4}$ out of $t^{4} (- t^{2}) + t^{4} (24 t)$ .

$f (t) = - 12 t^{5} - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t) + t^{4} \cdot - 32}{5}$

Step 1.13.2.5

Factor $t^{4}$ out of $t^{4} (- t^{2} + 24 t) + t^{4} \cdot - 32$ .

$f (t) = - 12 t^{5} - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{5 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.14

To write $- 12 t^{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - 12 t^{5} \cdot \frac{2}{2} + \frac{5 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.15

Simplify terms.

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Step 1.15.1

Combine $- 12 t^{5}$ and $\frac{2}{2}$ .

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{- 12 t^{5} \cdot 2}{2} + \frac{5 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.15.2

Combine the numerators over the common denominator.

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{- 12 t^{5} \cdot 2 + 5 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.16

Simplify each term.

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Step 1.16.1

Simplify the numerator.

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Step 1.16.1.1

Factor $t^{5}$ out of $- 12 t^{5} \cdot 2 + 5 t^{5}$ .

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Step 1.16.1.1.1

Factor $t^{5}$ out of $- 12 t^{5} \cdot 2$ .

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{t^{5} (- 12 \cdot 2) + 5 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.16.1.1.2

Factor $t^{5}$ out of $5 t^{5}$ .

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{t^{5} (- 12 \cdot 2) + t^{5} \cdot 5}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.16.1.1.3

Factor $t^{5}$ out of $t^{5} (- 12 \cdot 2) + t^{5} \cdot 5$ .

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{t^{5} (- 12 \cdot 2 + 5)}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.16.1.2

Multiply $- 12$ by $2$ .

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{t^{5} (- 24 + 5)}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.16.1.3

Add $- 24$ and $5$ .

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{t^{5} \cdot - 19}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.16.2

Move $- 19$ to the left of $t^{5}$ .

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} + \frac{- 19 \cdot t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.16.3

Move the negative in front of the fraction.

$f (t) = - 30 t^{4} + 48 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.17

Add $- 30 t^{4}$ and $48 t^{4}$ .

$f (t) = 18 t^{4} + 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.18

To write $18 t^{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (t) = 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + 18 t^{4} \cdot \frac{5}{5} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.19

Simplify terms.

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Step 1.19.1

Combine $18 t^{4}$ and $\frac{5}{5}$ .

$f (t) = 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{18 t^{4} \cdot 5}{5} + \frac{t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.19.2

Combine the numerators over the common denominator.

$f (t) = 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{18 t^{4} \cdot 5 + t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.20

Simplify the numerator.

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Step 1.20.1

Factor $t^{4}$ out of $18 t^{4} \cdot 5 + t^{4} (- t^{2} + 24 t - 32)$ .

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Step 1.20.1.1

Factor $t^{4}$ out of $18 t^{4} \cdot 5$ .

$f (t) = 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{4} (18 \cdot 5) + t^{4} (- t^{2} + 24 t - 32)}{5}$

Step 1.20.1.2

Factor $t^{4}$ out of $t^{4} (18 \cdot 5) + t^{4} (- t^{2} + 24 t - 32)$ .

$f (t) = 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{4} (18 \cdot 5 - t^{2} + 24 t - 32)}{5}$

Step 1.20.2

Multiply $18$ by $5$ .

$f (t) = 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{4} (90 - t^{2} + 24 t - 32)}{5}$

Step 1.20.3

Subtract $32$ from $90$ .

$f (t) = 120 t^{3} - 64 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t + 58)}{5}$

Step 1.21

Subtract $64 t^{3}$ from $120 t^{3}$ .

$f (t) = 56 t^{3} - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{4} (- t^{2} + 24 t + 58)}{5}$

Step 1.22

To write $56 t^{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + 56 t^{3} \cdot \frac{5}{5} + \frac{t^{4} (- t^{2} + 24 t + 58)}{5}$

Step 1.23

Simplify terms.

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Step 1.23.1

Combine $56 t^{3}$ and $\frac{5}{5}$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{56 t^{3} \cdot 5}{5} + \frac{t^{4} (- t^{2} + 24 t + 58)}{5}$

Step 1.23.2

Combine the numerators over the common denominator.

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{56 t^{3} \cdot 5 + t^{4} (- t^{2} + 24 t + 58)}{5}$

Step 1.24

Simplify the numerator.

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Step 1.24.1

Factor $t^{3}$ out of $56 t^{3} \cdot 5 + t^{4} (- t^{2} + 24 t + 58)$ .

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Step 1.24.1.1

Factor $t^{3}$ out of $56 t^{3} \cdot 5$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (56 \cdot 5) + t^{4} (- t^{2} + 24 t + 58)}{5}$

Step 1.24.1.2

Factor $t^{3}$ out of $t^{4} (- t^{2} + 24 t + 58)$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (56 \cdot 5) + t^{3} (t (- t^{2} + 24 t + 58))}{5}$

Step 1.24.1.3

Factor $t^{3}$ out of $t^{3} (56 \cdot 5) + t^{3} (t (- t^{2} + 24 t + 58))$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (56 \cdot 5 + t (- t^{2} + 24 t + 58))}{5}$

Step 1.24.2

Multiply $56$ by $5$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 + t (- t^{2} + 24 t + 58))}{5}$

Step 1.24.3

Apply the distributive property.

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 + t (- t^{2}) + t (24 t) + t \cdot 58)}{5}$

Step 1.24.4

Simplify.

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Step 1.24.4.1

Rewrite using the commutative property of multiplication.

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - t \cdot t^{2} + t (24 t) + t \cdot 58)}{5}$

Step 1.24.4.2

Rewrite using the commutative property of multiplication.

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - t \cdot t^{2} + 24 t \cdot t + t \cdot 58)}{5}$

Step 1.24.4.3

Move $58$ to the left of $t$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - t \cdot t^{2} + 24 t \cdot t + 58 \cdot t)}{5}$

Step 1.24.5

Simplify each term.

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Step 1.24.5.1

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

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Step 1.24.5.1.1

Move $t^{2}$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - (t^{2} t) + 24 t \cdot t + 58 \cdot t)}{5}$

Step 1.24.5.1.2

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

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Step 1.24.5.1.2.1

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

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - (t^{2} t) + 24 t \cdot t + 58 \cdot t)}{5}$

Step 1.24.5.1.2.2

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

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - t^{2 + 1} + 24 t \cdot t + 58 \cdot t)}{5}$

Step 1.24.5.1.3

Add $2$ and $1$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - t^{3} + 24 t \cdot t + 58 \cdot t)}{5}$

Step 1.24.5.2

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

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Step 1.24.5.2.1

Move $t$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - t^{3} + 24 (t \cdot t) + 58 \cdot t)}{5}$

Step 1.24.5.2.2

Multiply $t$ by $t$ .

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - t^{3} + 24 t^{2} + 58 \cdot t)}{5}$

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (280 - t^{3} + 24 t^{2} + 58 t)}{5}$

Step 1.24.6

Reorder terms.

$f (t) = - 160 t^{2} + \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{3} (- t^{3} + 24 t^{2} + 58 t + 280)}{5}$

Step 1.25

To write $- 160 t^{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} - 160 t^{2} \cdot \frac{5}{5} + \frac{t^{3} (- t^{3} + 24 t^{2} + 58 t + 280)}{5}$

Step 1.26

Simplify terms.

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Step 1.26.1

Combine $- 160 t^{2}$ and $\frac{5}{5}$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{- 160 t^{2} \cdot 5}{5} + \frac{t^{3} (- t^{3} + 24 t^{2} + 58 t + 280)}{5}$

Step 1.26.2

Combine the numerators over the common denominator.

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{- 160 t^{2} \cdot 5 + t^{3} (- t^{3} + 24 t^{2} + 58 t + 280)}{5}$

Step 1.27

Simplify the numerator.

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Step 1.27.1

Factor $t^{2}$ out of $- 160 t^{2} \cdot 5 + t^{3} (- t^{3} + 24 t^{2} + 58 t + 280)$ .

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Step 1.27.1.1

Factor $t^{2}$ out of $- 160 t^{2} \cdot 5$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 160 \cdot 5) + t^{3} (- t^{3} + 24 t^{2} + 58 t + 280)}{5}$

Step 1.27.1.2

Factor $t^{2}$ out of $t^{3} (- t^{3} + 24 t^{2} + 58 t + 280)$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 160 \cdot 5) + t^{2} (t (- t^{3} + 24 t^{2} + 58 t + 280))}{5}$

Step 1.27.1.3

Factor $t^{2}$ out of $t^{2} (- 160 \cdot 5) + t^{2} (t (- t^{3} + 24 t^{2} + 58 t + 280))$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 160 \cdot 5 + t (- t^{3} + 24 t^{2} + 58 t + 280))}{5}$

Step 1.27.2

Multiply $- 160$ by $5$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 + t (- t^{3} + 24 t^{2} + 58 t + 280))}{5}$

Step 1.27.3

Apply the distributive property.

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 + t (- t^{3}) + t (24 t^{2}) + t (58 t) + t \cdot 280)}{5}$

Step 1.27.4

Simplify.

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Step 1.27.4.1

Rewrite using the commutative property of multiplication.

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t \cdot t^{3} + t (24 t^{2}) + t (58 t) + t \cdot 280)}{5}$

Step 1.27.4.2

Rewrite using the commutative property of multiplication.

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t \cdot t^{3} + 24 t \cdot t^{2} + t (58 t) + t \cdot 280)}{5}$

Step 1.27.4.3

Rewrite using the commutative property of multiplication.

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t \cdot t^{3} + 24 t \cdot t^{2} + 58 t \cdot t + t \cdot 280)}{5}$

Step 1.27.4.4

Move $280$ to the left of $t$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t \cdot t^{3} + 24 t \cdot t^{2} + 58 t \cdot t + 280 \cdot t)}{5}$

Step 1.27.5

Simplify each term.

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Step 1.27.5.1

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

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Step 1.27.5.1.1

Move $t^{3}$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - (t^{3} t) + 24 t \cdot t^{2} + 58 t \cdot t + 280 \cdot t)}{5}$

Step 1.27.5.1.2

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

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Step 1.27.5.1.2.1

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

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - (t^{3} t) + 24 t \cdot t^{2} + 58 t \cdot t + 280 \cdot t)}{5}$

Step 1.27.5.1.2.2

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

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t^{3 + 1} + 24 t \cdot t^{2} + 58 t \cdot t + 280 \cdot t)}{5}$

Step 1.27.5.1.3

Add $3$ and $1$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t^{4} + 24 t \cdot t^{2} + 58 t \cdot t + 280 \cdot t)}{5}$

Step 1.27.5.2

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

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Step 1.27.5.2.1

Move $t^{2}$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t^{4} + 24 (t^{2} t) + 58 t \cdot t + 280 \cdot t)}{5}$

Step 1.27.5.2.2

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

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Step 1.27.5.2.2.1

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

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t^{4} + 24 (t^{2} t) + 58 t \cdot t + 280 \cdot t)}{5}$

Step 1.27.5.2.2.2

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

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t^{4} + 24 t^{2 + 1} + 58 t \cdot t + 280 \cdot t)}{5}$

Step 1.27.5.2.3

Add $2$ and $1$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t^{4} + 24 t^{3} + 58 t \cdot t + 280 \cdot t)}{5}$

Step 1.27.5.3

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

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Step 1.27.5.3.1

Move $t$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t^{4} + 24 t^{3} + 58 (t \cdot t) + 280 \cdot t)}{5}$

Step 1.27.5.3.2

Multiply $t$ by $t$ .

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t^{4} + 24 t^{3} + 58 t^{2} + 280 \cdot t)}{5}$

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- 800 - t^{4} + 24 t^{3} + 58 t^{2} + 280 t)}{5}$

Step 1.27.6

Reorder terms.

$f (t) = \frac{t^{7}}{10} - \frac{19 t^{5}}{2} + \frac{t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800)}{5}$

Step 1.28

To write $\frac{t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800)}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{7}}{10} + \frac{t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800)}{5} \cdot \frac{2}{2}$

Step 1.29

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.29.1

Multiply $\frac{t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800)}{5}$ by $\frac{2}{2}$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{7}}{10} + \frac{t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800) \cdot 2}{5 \cdot 2}$

Step 1.29.2

Multiply $5$ by $2$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{7}}{10} + \frac{t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800) \cdot 2}{10}$

Step 1.30

Combine the numerators over the common denominator.

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{7} + t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800) \cdot 2}{10}$

Step 1.31

Simplify the numerator.

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Step 1.31.1

Factor $t^{2}$ out of $t^{7} + t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800) \cdot 2$ .

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Step 1.31.1.1

Factor $t^{2}$ out of $t^{7}$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{2} t^{5} + t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800) \cdot 2}{10}$

Step 1.31.1.2

Factor $t^{2}$ out of $t^{2} (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800) \cdot 2$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{2} t^{5} + t^{2} ((- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800) \cdot 2)}{10}$

Step 1.31.1.3

Factor $t^{2}$ out of $t^{2} t^{5} + t^{2} ((- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800) \cdot 2)$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{2} (t^{5} + (- t^{4} + 24 t^{3} + 58 t^{2} + 280 t - 800) \cdot 2)}{10}$

Step 1.31.2

Apply the distributive property.

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{2} (t^{5} - t^{4} \cdot 2 + 24 t^{3} \cdot 2 + 58 t^{2} \cdot 2 + 280 t \cdot 2 - 800 \cdot 2)}{10}$

Step 1.31.3

Simplify.

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Step 1.31.3.1

Multiply $2$ by $- 1$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{2} (t^{5} - 2 t^{4} + 24 t^{3} \cdot 2 + 58 t^{2} \cdot 2 + 280 t \cdot 2 - 800 \cdot 2)}{10}$

Step 1.31.3.2

Multiply $2$ by $24$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 58 t^{2} \cdot 2 + 280 t \cdot 2 - 800 \cdot 2)}{10}$

Step 1.31.3.3

Multiply $2$ by $58$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 280 t \cdot 2 - 800 \cdot 2)}{10}$

Step 1.31.3.4

Multiply $2$ by $280$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 800 \cdot 2)}{10}$

Step 1.31.3.5

Multiply $- 800$ by $2$ .

$f (t) = - \frac{19 t^{5}}{2} + \frac{t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 1.32

To write $- \frac{19 t^{5}}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (t) = - \frac{19 t^{5}}{2} \cdot \frac{5}{5} + \frac{t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 1.33

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.33.1

Multiply $\frac{19 t^{5}}{2}$ by $\frac{5}{5}$ .

$f (t) = - \frac{19 t^{5} \cdot 5}{2 \cdot 5} + \frac{t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 1.33.2

Multiply $2$ by $5$ .

$f (t) = - \frac{19 t^{5} \cdot 5}{10} + \frac{t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 1.34

Combine the numerators over the common denominator.

$f (t) = \frac{- 19 t^{5} \cdot 5 + t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 1.35

Simplify the numerator.

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Step 1.35.1

Factor $t^{2}$ out of $- 19 t^{5} \cdot 5 + t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)$ .

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Step 1.35.1.1

Factor $t^{2}$ out of $- 19 t^{5} \cdot 5$ .

$f (t) = \frac{t^{2} (- 19 t^{3} \cdot 5) + t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 1.35.1.2

Factor $t^{2}$ out of $t^{2} (- 19 t^{3} \cdot 5) + t^{2} (t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)$ .

$f (t) = \frac{t^{2} (- 19 t^{3} \cdot 5 + t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 1.35.2

Multiply $5$ by $- 19$ .

$f (t) = \frac{t^{2} (- 95 t^{3} + t^{5} - 2 t^{4} + 48 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 1.35.3

Add $- 95 t^{3}$ and $48 t^{3}$ .

$f (t) = \frac{t^{2} (t^{5} - 2 t^{4} - 47 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 2

The largest exponent is the degree of the polynomial.

$0$

Step 3

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

$\frac{t^{2} (t^{5} - 2 t^{4} - 47 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

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.

$\frac{t^{2} (t^{5} - 2 t^{4} - 47 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Step 4.2

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

$\frac{1}{10}$

Step 5

List the results.

Polynomial Degree: $0$

Leading Term: $\frac{t^{2} (t^{5} - 2 t^{4} - 47 t^{3} + 116 t^{2} + 560 t - 1600)}{10}$

Leading Coefficient: $\frac{1}{10}$