Calculus Examples

$g (t) = \frac{1 + t + t^{2}}{t - t^{3}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = 1 + t + t^{2}$ and $g (t) = t - t^{3}$ .

$\frac{(t - t^{3}) \frac{d}{d t} [1 + t + t^{2}] - (1 + t + t^{2}) \frac{d}{d t} [t - t^{3}]}{{(t - t^{3})}^{2}}$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $1 + t + t^{2}$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [t] + \frac{d}{d t} [t^{2}]$ .

$\frac{(t - t^{3}) (\frac{d}{d t} [1] + \frac{d}{d t} [t] + \frac{d}{d t} [t^{2}]) - (1 + t + t^{2}) \frac{d}{d t} [t - t^{3}]}{{(t - t^{3})}^{2}}$

Step 2.2

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$\frac{(t - t^{3}) (0 + \frac{d}{d t} [t] + \frac{d}{d t} [t^{2}]) - (1 + t + t^{2}) \frac{d}{d t} [t - t^{3}]}{{(t - t^{3})}^{2}}$

Step 2.3

Add $0$ and $\frac{d}{d t} [t]$ .

$\frac{(t - t^{3}) (\frac{d}{d t} [t] + \frac{d}{d t} [t^{2}]) - (1 + t + t^{2}) \frac{d}{d t} [t - t^{3}]}{{(t - t^{3})}^{2}}$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{(t - t^{3}) (1 + \frac{d}{d t} [t^{2}]) - (1 + t + t^{2}) \frac{d}{d t} [t - t^{3}]}{{(t - t^{3})}^{2}}$

Step 2.5

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$\frac{(t - t^{3}) (1 + 2 t) - (1 + t + t^{2}) \frac{d}{d t} [t - t^{3}]}{{(t - t^{3})}^{2}}$

Step 2.6

By the Sum Rule, the derivative of $t - t^{3}$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- t^{3}]$ .

$\frac{(t - t^{3}) (1 + 2 t) - (1 + t + t^{2}) (\frac{d}{d t} [t] + \frac{d}{d t} [- t^{3}])}{{(t - t^{3})}^{2}}$

Step 2.7

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{(t - t^{3}) (1 + 2 t) - (1 + t + t^{2}) (1 + \frac{d}{d t} [- t^{3}])}{{(t - t^{3})}^{2}}$

Step 2.8

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{3}$ with respect to $t$ is $- \frac{d}{d t} [t^{3}]$ .

$\frac{(t - t^{3}) (1 + 2 t) - (1 + t + t^{2}) (1 - \frac{d}{d t} [t^{3}])}{{(t - t^{3})}^{2}}$

Step 2.9

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$\frac{(t - t^{3}) (1 + 2 t) - (1 + t + t^{2}) (1 - (3 t^{2}))}{{(t - t^{3})}^{2}}$

Step 2.10

Multiply $3$ by $- 1$ .

$\frac{(t - t^{3}) (1 + 2 t) - (1 + t + t^{2}) (1 - 3 t^{2})}{{(t - t^{3})}^{2}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify the numerator.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Expand $(t - t^{3}) (1 + 2 t)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.1.1

Apply the distributive property.

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

Step 3.2.1.1.2

Apply the distributive property.

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

Step 3.2.1.1.3

Apply the distributive property.

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

Step 3.2.1.2

Simplify each term.

Tap for more steps...

Step 3.2.1.2.1

Multiply $t$ by $1$ .

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

Step 3.2.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.3

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

Tap for more steps...

Step 3.2.1.2.3.1

Move $t$ .

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

Step 3.2.1.2.3.2

Multiply $t$ by $t$ .

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

Step 3.2.1.2.4

Multiply $- 1$ by $1$ .

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

Step 3.2.1.2.5

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.6

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

Tap for more steps...

Step 3.2.1.2.6.1

Move $t$ .

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

Step 3.2.1.2.6.2

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

Tap for more steps...

Step 3.2.1.2.6.2.1

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

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

Step 3.2.1.2.6.2.2

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

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

Step 3.2.1.2.6.3

Add $1$ and $3$ .

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

Step 3.2.1.2.7

Multiply $- 1$ by $2$ .

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

Step 3.2.1.3

Multiply $- 1$ by $1$ .

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

Step 3.2.1.4

Expand $(- 1 - t - t^{2}) (1 - 3 t^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.2.1.5

Simplify each term.

Tap for more steps...

Step 3.2.1.5.1

Multiply $- 1$ by $1$ .

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

Step 3.2.1.5.2

Multiply $- 3$ by $- 1$ .

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

Step 3.2.1.5.3

Multiply $- 1$ by $1$ .

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

Step 3.2.1.5.4

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.5.5

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

Tap for more steps...

Step 3.2.1.5.5.1

Move $t^{2}$ .

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

Step 3.2.1.5.5.2

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

Tap for more steps...

Step 3.2.1.5.5.2.1

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

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

Step 3.2.1.5.5.2.2

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

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

Step 3.2.1.5.5.3

Add $2$ and $1$ .

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

Step 3.2.1.5.6

Multiply $- 1$ by $- 3$ .

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

Step 3.2.1.5.7

Multiply $- 1$ by $1$ .

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

Step 3.2.1.5.8

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.5.9

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

Tap for more steps...

Step 3.2.1.5.9.1

Move $t^{2}$ .

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

Step 3.2.1.5.9.2

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

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

Step 3.2.1.5.9.3

Add $2$ and $2$ .

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

Step 3.2.1.5.10

Multiply $- 1$ by $- 3$ .

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

Step 3.2.1.6

Subtract $t^{2}$ from $3 t^{2}$ .

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

Step 3.2.2

Combine the opposite terms in $t + 2 t^{2} - t^{3} - 2 t^{4} - 1 + 2 t^{2} - t + 3 t^{3} + 3 t^{4}$ .

Tap for more steps...

Step 3.2.2.1

Subtract $t$ from $t$ .

$\frac{2 t^{2} - t^{3} - 2 t^{4} - 1 + 2 t^{2} + 0 + 3 t^{3} + 3 t^{4}}{{(t - t^{3})}^{2}}$

Step 3.2.2.2

Add $2 t^{2} - t^{3} - 2 t^{4} - 1 + 2 t^{2}$ and $0$ .

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Add $- 2 t^{4}$ and $3 t^{4}$ .

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

Step 3.3

Reorder terms.

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

Step 3.4

Simplify the denominator.

Tap for more steps...

Step 3.4.1

Factor $t$ out of $- t^{3} + t$ .

Tap for more steps...

Step 3.4.1.1

Factor $t$ out of $- t^{3}$ .

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

Step 3.4.1.2

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

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

Step 3.4.1.3

Factor $t$ out of $t^{1}$ .

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

Step 3.4.1.4

Factor $t$ out of $t (- t^{2}) + t \cdot 1$ .

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

Step 3.4.2

Rewrite $1$ as $1^{2}$ .

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

Step 3.4.3

Reorder $- t^{2}$ and $1^{2}$ .

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

Step 3.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = t$ .

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

Step 3.4.5

Apply the product rule to $t (1 + t) (1 - t)$ .

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

Step 3.4.6

Apply the distributive property.

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

Step 3.4.7

Multiply $t$ by $1$ .

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

Step 3.4.8

Multiply $t$ by $t$ .

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

Step 3.4.9

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

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

Step 3.4.10

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

Tap for more steps...

Step 3.4.10.1

Apply the distributive property.

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

Step 3.4.10.2

Apply the distributive property.

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

Step 3.4.10.3

Apply the distributive property.

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

Step 3.4.11

Simplify and combine like terms.

Tap for more steps...

Step 3.4.11.1

Simplify each term.

Tap for more steps...

Step 3.4.11.1.1

Multiply $t$ by $t$ .

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

Step 3.4.11.1.2

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

Tap for more steps...

Step 3.4.11.1.2.1

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

Tap for more steps...

Step 3.4.11.1.2.1.1

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

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

Step 3.4.11.1.2.1.2

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

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

Step 3.4.11.1.2.2

Add $1$ and $2$ .

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

Step 3.4.11.1.3

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

Tap for more steps...

Step 3.4.11.1.3.1

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

Tap for more steps...

Step 3.4.11.1.3.1.1

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

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

Step 3.4.11.1.3.1.2

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

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

Step 3.4.11.1.3.2

Add $2$ and $1$ .

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

Step 3.4.11.1.4

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

Tap for more steps...

Step 3.4.11.1.4.1

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

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

Step 3.4.11.1.4.2

Add $2$ and $2$ .

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

Step 3.4.11.2

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

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

Step 3.4.12

Factor $t^{2}$ out of $t^{2} + 2 t^{3} + t^{4}$ .

Tap for more steps...

Step 3.4.12.1

Multiply by $1$ .

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

Step 3.4.12.2

Factor $t^{2}$ out of $2 t^{3}$ .

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

Step 3.4.12.3

Factor $t^{2}$ out of $t^{4}$ .

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

Step 3.4.12.4

Factor $t^{2}$ out of $t^{2} \cdot 1 + t^{2} (2 t)$ .

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

Step 3.4.12.5

Factor $t^{2}$ out of $t^{2} \cdot (1 + 2 t) + t^{2} t^{2}$ .

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

Step 3.4.13

Factor using the perfect square rule.

Tap for more steps...

Step 3.4.13.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.4.13.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 t = 2 \cdot 1 \cdot t$

Step 3.4.13.3

Rewrite the polynomial.

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

Step 3.4.13.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 1$ and $b = t$ .

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