Calculus Examples

$Q (r) = (r^{2} + 1) (r^{3} - r) (3 r^{4} + 2 r - 1)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d r} [f (r) g (r)]$ is $f (r) \frac{d}{d r} [g (r)] + g (r) \frac{d}{d r} [f (r)]$ where $f (r) = (r^{2} + 1) (r^{3} - r)$ and $g (r) = 3 r^{4} + 2 r - 1$ .

$(r^{2} + 1) (r^{3} - r) \frac{d}{d r} [3 r^{4} + 2 r - 1] + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 r^{4} + 2 r - 1$ with respect to $r$ is $\frac{d}{d r} [3 r^{4}] + \frac{d}{d r} [2 r] + \frac{d}{d r} [- 1]$ .

$(r^{2} + 1) (r^{3} - r) (\frac{d}{d r} [3 r^{4}] + \frac{d}{d r} [2 r] + \frac{d}{d r} [- 1]) + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 2.2

Since $3$ is constant with respect to $r$ , the derivative of $3 r^{4}$ with respect to $r$ is $3 \frac{d}{d r} [r^{4}]$ .

$(r^{2} + 1) (r^{3} - r) (3 \frac{d}{d r} [r^{4}] + \frac{d}{d r} [2 r] + \frac{d}{d r} [- 1]) + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = 4$ .

$(r^{2} + 1) (r^{3} - r) (3 (4 r^{3}) + \frac{d}{d r} [2 r] + \frac{d}{d r} [- 1]) + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 2.4

Multiply $4$ by $3$ .

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + \frac{d}{d r} [2 r] + \frac{d}{d r} [- 1]) + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 2.5

Since $2$ is constant with respect to $r$ , the derivative of $2 r$ with respect to $r$ is $2 \frac{d}{d r} [r]$ .

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2 \frac{d}{d r} [r] + \frac{d}{d r} [- 1]) + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 2.6

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2 \cdot 1 + \frac{d}{d r} [- 1]) + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 2.7

Multiply $2$ by $1$ .

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2 + \frac{d}{d r} [- 1]) + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 2.8

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2 + 0) + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 2.9

Add $12 r^{3} + 2$ and $0$ .

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) \frac{d}{d r} [(r^{2} + 1) (r^{3} - r)]$

Step 3

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) \frac{d}{d r} [r^{3} - r] + (r^{3} - r) \frac{d}{d r} [r^{2} + 1])$

Step 4

Differentiate.

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Step 4.1

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (\frac{d}{d r} [r^{3}] + \frac{d}{d r} [- r]) + (r^{3} - r) \frac{d}{d r} [r^{2} + 1])$

Step 4.2

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} + \frac{d}{d r} [- r]) + (r^{3} - r) \frac{d}{d r} [r^{2} + 1])$

Step 4.3

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - \frac{d}{d r} [r]) + (r^{3} - r) \frac{d}{d r} [r^{2} + 1])$

Step 4.4

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1 \cdot 1) + (r^{3} - r) \frac{d}{d r} [r^{2} + 1])$

Step 4.5

Multiply $- 1$ by $1$ .

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + (r^{3} - r) \frac{d}{d r} [r^{2} + 1])$

Step 4.6

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + (r^{3} - r) (\frac{d}{d r} [r^{2}] + \frac{d}{d r} [1]))$

Step 4.7

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + (r^{3} - r) (2 r + \frac{d}{d r} [1]))$

Step 4.8

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + (r^{3} - r) (2 r + 0))$

Step 4.9

Simplify the expression.

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Step 4.9.1

Add $2 r$ and $0$ .

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + (r^{3} - r) (2 r))$

Step 4.9.2

Move $2$ to the left of $r^{3} - r$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + (2 r^{3} + 2 (- r)) r)$

Step 5.2

Apply the distributive property.

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{3} r + 2 (- r) r)$

Step 5.3

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 (r^{1} r^{3}) + 2 (- r) r)$

Step 5.4

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{1 + 3} + 2 (- r) r)$

Step 5.5

Add $1$ and $3$ .

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} + 2 (- r) r)$

Step 5.6

Multiply $- 1$ by $2$ .

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

Step 5.7

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 (r^{1} r))$

Step 5.8

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 (r^{1} r^{1}))$

Step 5.9

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

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{1 + 1})$

Step 5.10

Add $1$ and $1$ .

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2})$

Step 5.11

Factor $1$ out of $(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2})$ .

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Step 5.11.1

Factor $1$ out of $(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2)$ .

$1 (((r^{2} + 1) (r^{3} - r)) (12 r^{3} + 2)) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2})$

Step 5.11.2

Factor $1$ out of $(3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2})$ .

$1 (((r^{2} + 1) (r^{3} - r)) (12 r^{3} + 2)) + 1 ((3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}))$

Step 5.11.3

Factor $1$ out of $1 (((r^{2} + 1) (r^{3} - r)) (12 r^{3} + 2)) + 1 ((3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}))$ .

$1 (((r^{2} + 1) (r^{3} - r)) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}))$

Step 5.12

Multiply $(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2})$ by $1$ .

$(r^{2} + 1) (r^{3} - r) (12 r^{3} + 2) + (3 r^{4} + 2 r - 1) ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2})$

Step 5.13

Reorder terms.

$(12 r^{3} + 2) (r^{2} + 1) (r^{3} - r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14

Simplify each term.

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Step 5.14.1

Expand $(12 r^{3} + 2) (r^{2} + 1)$ using the FOIL Method.

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Step 5.14.1.1

Apply the distributive property.

$(12 r^{3} (r^{2} + 1) + 2 (r^{2} + 1)) (r^{3} - r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.1.2

Apply the distributive property.

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

Step 5.14.1.3

Apply the distributive property.

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

Step 5.14.2

Simplify each term.

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Step 5.14.2.1

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

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Step 5.14.2.1.1

Move $r^{2}$ .

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

Step 5.14.2.1.2

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

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

Step 5.14.2.1.3

Add $2$ and $3$ .

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

Step 5.14.2.2

Multiply $12$ by $1$ .

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

Step 5.14.2.3

Multiply $2$ by $1$ .

$(12 r^{5} + 12 r^{3} + 2 r^{2} + 2) (r^{3} - r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.3

Expand $(12 r^{5} + 12 r^{3} + 2 r^{2} + 2) (r^{3} - r)$ by multiplying each term in the first expression by each term in the second expression.

$12 r^{5} r^{3} + 12 r^{5} (- r) + 12 r^{3} r^{3} + 12 r^{3} (- r) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4

Simplify each term.

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Step 5.14.4.1

Multiply $r^{5}$ by $r^{3}$ by adding the exponents.

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Step 5.14.4.1.1

Move $r^{3}$ .

$12 (r^{3} r^{5}) + 12 r^{5} (- r) + 12 r^{3} r^{3} + 12 r^{3} (- r) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.1.2

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

$12 r^{3 + 5} + 12 r^{5} (- r) + 12 r^{3} r^{3} + 12 r^{3} (- r) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.1.3

Add $3$ and $5$ .

$12 r^{8} + 12 r^{5} (- r) + 12 r^{3} r^{3} + 12 r^{3} (- r) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.2

Rewrite using the commutative property of multiplication.

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

Step 5.14.4.3

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

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Step 5.14.4.3.1

Move $r$ .

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

Step 5.14.4.3.2

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

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Step 5.14.4.3.2.1

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

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

Step 5.14.4.3.2.2

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

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

Step 5.14.4.3.3

Add $1$ and $5$ .

$12 r^{8} + 12 \cdot - 1 r^{6} + 12 r^{3} r^{3} + 12 r^{3} (- r) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.4

Multiply $12$ by $- 1$ .

$12 r^{8} - 12 r^{6} + 12 r^{3} r^{3} + 12 r^{3} (- r) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.5

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

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Step 5.14.4.5.1

Move $r^{3}$ .

$12 r^{8} - 12 r^{6} + 12 (r^{3} r^{3}) + 12 r^{3} (- r) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.5.2

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

$12 r^{8} - 12 r^{6} + 12 r^{3 + 3} + 12 r^{3} (- r) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.5.3

Add $3$ and $3$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} + 12 r^{3} (- r) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.6

Rewrite using the commutative property of multiplication.

$12 r^{8} - 12 r^{6} + 12 r^{6} + 12 \cdot - 1 r^{3} r + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.7

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

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Step 5.14.4.7.1

Move $r$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} + 12 \cdot - 1 (r \cdot r^{3}) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.7.2

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

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Step 5.14.4.7.2.1

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

$12 r^{8} - 12 r^{6} + 12 r^{6} + 12 \cdot - 1 (r^{1} r^{3}) + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.7.2.2

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

$12 r^{8} - 12 r^{6} + 12 r^{6} + 12 \cdot - 1 r^{1 + 3} + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.7.3

Add $1$ and $3$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} + 12 \cdot - 1 r^{4} + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.8

Multiply $12$ by $- 1$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{2} r^{3} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.9

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

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Step 5.14.4.9.1

Move $r^{3}$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 (r^{3} r^{2}) + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.9.2

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

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{3 + 2} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.9.3

Add $3$ and $2$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{5} + 2 r^{2} (- r) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.10

Rewrite using the commutative property of multiplication.

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{5} + 2 \cdot - 1 r^{2} r + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.11

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

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Step 5.14.4.11.1

Move $r$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{5} + 2 \cdot - 1 (r \cdot r^{2}) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.11.2

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

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Step 5.14.4.11.2.1

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

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{5} + 2 \cdot - 1 (r^{1} r^{2}) + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.11.2.2

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

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{5} + 2 \cdot - 1 r^{1 + 2} + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.11.3

Add $1$ and $2$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{5} + 2 \cdot - 1 r^{3} + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.12

Multiply $2$ by $- 1$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{5} - 2 r^{3} + 2 r^{3} + 2 (- r) + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.4.13

Multiply $- 1$ by $2$ .

$12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{5} - 2 r^{3} + 2 r^{3} - 2 r + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.5

Combine the opposite terms in $12 r^{8} - 12 r^{6} + 12 r^{6} - 12 r^{4} + 2 r^{5} - 2 r^{3} + 2 r^{3} - 2 r$ .

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Step 5.14.5.1

Add $- 12 r^{6}$ and $12 r^{6}$ .

$12 r^{8} + 0 - 12 r^{4} + 2 r^{5} - 2 r^{3} + 2 r^{3} - 2 r + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.5.2

Add $12 r^{8}$ and $0$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r^{3} + 2 r^{3} - 2 r + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.5.3

Add $- 2 r^{3}$ and $2 r^{3}$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} + 0 - 2 r + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.5.4

Add $12 r^{8} - 12 r^{4} + 2 r^{5}$ and $0$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + ((r^{2} + 1) (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.6

Simplify each term.

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Step 5.14.6.1

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

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Step 5.14.6.1.1

Apply the distributive property.

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + (r^{2} (3 r^{2} - 1) + 1 (3 r^{2} - 1) + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.6.1.2

Apply the distributive property.

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

Step 5.14.6.1.3

Apply the distributive property.

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

Step 5.14.6.2

Simplify and combine like terms.

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Step 5.14.6.2.1

Simplify each term.

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Step 5.14.6.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.14.6.2.1.2

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

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Step 5.14.6.2.1.2.1

Move $r^{2}$ .

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

Step 5.14.6.2.1.2.2

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

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

Step 5.14.6.2.1.2.3

Add $2$ and $2$ .

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

Step 5.14.6.2.1.3

Move $- 1$ to the left of $r^{2}$ .

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

Step 5.14.6.2.1.4

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

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

Step 5.14.6.2.1.5

Multiply $3 r^{2}$ by $1$ .

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

Step 5.14.6.2.1.6

Multiply $- 1$ by $1$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + (3 r^{4} - r^{2} + 3 r^{2} - 1 + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.6.2.2

Add $- r^{2}$ and $3 r^{2}$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + (3 r^{4} + 2 r^{2} - 1 + 2 r^{4} - 2 r^{2}) (3 r^{4} + 2 r - 1)$

Step 5.14.7

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

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Step 5.14.7.1

Subtract $2 r^{2}$ from $2 r^{2}$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + (3 r^{4} - 1 + 2 r^{4} + 0) (3 r^{4} + 2 r - 1)$

Step 5.14.7.2

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

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + (3 r^{4} - 1 + 2 r^{4}) (3 r^{4} + 2 r - 1)$

Step 5.14.8

Add $3 r^{4}$ and $2 r^{4}$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + (5 r^{4} - 1) (3 r^{4} + 2 r - 1)$

Step 5.14.9

Expand $(5 r^{4} - 1) (3 r^{4} + 2 r - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5.14.10

Simplify each term.

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Step 5.14.10.1

Rewrite using the commutative property of multiplication.

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

Step 5.14.10.2

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

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Step 5.14.10.2.1

Move $r^{4}$ .

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

Step 5.14.10.2.2

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

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

Step 5.14.10.2.3

Add $4$ and $4$ .

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

Step 5.14.10.3

Multiply $5$ by $3$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 5 r^{4} (2 r) + 5 r^{4} \cdot - 1 - 1 (3 r^{4}) - 1 (2 r) - 1 \cdot - 1$

Step 5.14.10.4

Rewrite using the commutative property of multiplication.

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 5 \cdot 2 r^{4} r + 5 r^{4} \cdot - 1 - 1 (3 r^{4}) - 1 (2 r) - 1 \cdot - 1$

Step 5.14.10.5

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

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Step 5.14.10.5.1

Move $r$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 5 \cdot 2 (r \cdot r^{4}) + 5 r^{4} \cdot - 1 - 1 (3 r^{4}) - 1 (2 r) - 1 \cdot - 1$

Step 5.14.10.5.2

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

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Step 5.14.10.5.2.1

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

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 5 \cdot 2 (r^{1} r^{4}) + 5 r^{4} \cdot - 1 - 1 (3 r^{4}) - 1 (2 r) - 1 \cdot - 1$

Step 5.14.10.5.2.2

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

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 5 \cdot 2 r^{1 + 4} + 5 r^{4} \cdot - 1 - 1 (3 r^{4}) - 1 (2 r) - 1 \cdot - 1$

Step 5.14.10.5.3

Add $1$ and $4$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 5 \cdot 2 r^{5} + 5 r^{4} \cdot - 1 - 1 (3 r^{4}) - 1 (2 r) - 1 \cdot - 1$

Step 5.14.10.6

Multiply $5$ by $2$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 10 r^{5} + 5 r^{4} \cdot - 1 - 1 (3 r^{4}) - 1 (2 r) - 1 \cdot - 1$

Step 5.14.10.7

Multiply $- 1$ by $5$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 10 r^{5} - 5 r^{4} - 1 (3 r^{4}) - 1 (2 r) - 1 \cdot - 1$

Step 5.14.10.8

Multiply $3$ by $- 1$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 10 r^{5} - 5 r^{4} - 3 r^{4} - 1 (2 r) - 1 \cdot - 1$

Step 5.14.10.9

Multiply $2$ by $- 1$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 10 r^{5} - 5 r^{4} - 3 r^{4} - 2 r - 1 \cdot - 1$

Step 5.14.10.10

Multiply $- 1$ by $- 1$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 10 r^{5} - 5 r^{4} - 3 r^{4} - 2 r + 1$

Step 5.14.11

Subtract $3 r^{4}$ from $- 5 r^{4}$ .

$12 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 15 r^{8} + 10 r^{5} - 8 r^{4} - 2 r + 1$

Step 5.15

Add $12 r^{8}$ and $15 r^{8}$ .

$27 r^{8} - 12 r^{4} + 2 r^{5} - 2 r + 10 r^{5} - 8 r^{4} - 2 r + 1$

Step 5.16

Subtract $8 r^{4}$ from $- 12 r^{4}$ .

$27 r^{8} + 2 r^{5} - 2 r + 10 r^{5} - 20 r^{4} - 2 r + 1$

Step 5.17

Add $2 r^{5}$ and $10 r^{5}$ .

$27 r^{8} + 12 r^{5} - 2 r - 20 r^{4} - 2 r + 1$

Step 5.18

Subtract $2 r$ from $- 2 r$ .

$27 r^{8} + 12 r^{5} - 20 r^{4} - 4 r + 1$