Precalculus Examples

$T (x) = x^{3} (7 - x^{2}) + 98.6$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$T (x + h) = {(x + h)}^{3} (7 - {(x + h)}^{2}) + 98.6$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Use the Binomial Theorem.

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - {(x + h)}^{2}) + 98.6$

Step 2.1.2.1.2

Simplify each term.

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Step 2.1.2.1.2.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - ((x + h) (x + h))) + 98.6$

Step 2.1.2.1.2.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.1.2.2.1

Apply the distributive property.

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - (x (x + h) + h (x + h))) + 98.6$

Step 2.1.2.1.2.2.2

Apply the distributive property.

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - (x \cdot x + x h + h (x + h))) + 98.6$

Step 2.1.2.1.2.2.3

Apply the distributive property.

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - (x \cdot x + x h + h x + h \cdot h)) + 98.6$

Step 2.1.2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.1.2.3.1

Simplify each term.

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Step 2.1.2.1.2.3.1.1

Multiply $x$ by $x$ .

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - (x^{2} + x h + h x + h \cdot h)) + 98.6$

Step 2.1.2.1.2.3.1.2

Multiply $h$ by $h$ .

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - (x^{2} + x h + h x + h^{2})) + 98.6$

Step 2.1.2.1.2.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.2.3.2.1

Reorder $x$ and $h$ .

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - (x^{2} + h x + h x + h^{2})) + 98.6$

Step 2.1.2.1.2.3.2.2

Add $h x$ and $h x$ .

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - (x^{2} + 2 h x + h^{2})) + 98.6$

Step 2.1.2.1.2.4

Apply the distributive property.

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - x^{2} - (2 h x) - h^{2}) + 98.6$

Step 2.1.2.1.2.5

Multiply $2$ by $- 1$ .

$T (x + h) = (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - x^{2} - 2 h x - h^{2}) + 98.6$

Step 2.1.2.1.3

Expand $(x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) (7 - x^{2} - 2 h x - h^{2})$ by multiplying each term in the first expression by each term in the second expression.

$T (x + h) = x^{3} \cdot 7 + x^{3} (- x^{2}) + x^{3} (- 2 h x) + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4

Simplify each term.

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Step 2.1.2.1.4.1

Move $7$ to the left of $x^{3}$ .

$T (x + h) = 7 \cdot x^{3} + x^{3} (- x^{2}) + x^{3} (- 2 h x) + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.2

Rewrite using the commutative property of multiplication.

$T (x + h) = 7 x^{3} - x^{3} x^{2} + x^{3} (- 2 h x) + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.3

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

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Step 2.1.2.1.4.3.1

Move $x^{2}$ .

$T (x + h) = 7 x^{3} - (x^{2} x^{3}) + x^{3} (- 2 h x) + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.3.2

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

$T (x + h) = 7 x^{3} - x^{2 + 3} + x^{3} (- 2 h x) + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.3.3

Add $2$ and $3$ .

$T (x + h) = 7 x^{3} - x^{5} + x^{3} (- 2 h x) + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.4

Rewrite using the commutative property of multiplication.

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{3} (h x) + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.5

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

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Step 2.1.2.1.4.5.1

Move $x$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 (x \cdot x^{3}) h + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.5.2

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

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Step 2.1.2.1.4.5.2.1

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

Step 2.1.2.1.4.5.2.2

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

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{1 + 3} h + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.5.3

Add $1$ and $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h + x^{3} (- h^{2}) + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.6

Rewrite using the commutative property of multiplication.

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 3 x^{2} h \cdot 7 + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.7

Multiply $7$ by $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h + 3 x^{2} h (- x^{2}) + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.8

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

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Step 2.1.2.1.4.8.1

Move $x^{2}$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h + 3 (x^{2} x^{2}) h \cdot - 1 + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.8.2

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

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h + 3 x^{2 + 2} h \cdot - 1 + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.8.3

Add $2$ and $2$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h + 3 x^{4} h \cdot - 1 + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.9

Multiply $- 1$ by $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h + 3 x^{2} h (- 2 h x) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.10

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

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Step 2.1.2.1.4.10.1

Move $x$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h + 3 (x \cdot x^{2}) h (- 2 h) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.10.2

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

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Step 2.1.2.1.4.10.2.1

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

Step 2.1.2.1.4.10.2.2

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

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h + 3 x^{1 + 2} h (- 2 h) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.10.3

Add $1$ and $2$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h + 3 x^{3} h (- 2 h) + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.11

Multiply $h$ by $h$ by adding the exponents.

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Step 2.1.2.1.4.11.1

Move $h$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h + 3 x^{3} (h \cdot h) \cdot - 2 + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.11.2

Multiply $h$ by $h$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h + 3 x^{3} h^{2} \cdot - 2 + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.12

Multiply $- 2$ by $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} + 3 x^{2} h (- h^{2}) + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.13

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

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Step 2.1.2.1.4.13.1

Move $h^{2}$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} + 3 x^{2} (h^{2} h) \cdot - 1 + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.13.2

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

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Step 2.1.2.1.4.13.2.1

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

Step 2.1.2.1.4.13.2.2

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

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} + 3 x^{2} h^{2 + 1} \cdot - 1 + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.13.3

Add $2$ and $1$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} + 3 x^{2} h^{3} \cdot - 1 + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.14

Multiply $- 1$ by $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 3 x h^{2} \cdot 7 + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.15

Multiply $7$ by $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} + 3 x h^{2} (- x^{2}) + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.16

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

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Step 2.1.2.1.4.16.1

Move $x^{2}$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} + 3 (x^{2} x) h^{2} \cdot - 1 + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.16.2

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

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Step 2.1.2.1.4.16.2.1

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

Step 2.1.2.1.4.16.2.2

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

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} + 3 x^{2 + 1} h^{2} \cdot - 1 + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.16.3

Add $2$ and $1$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} + 3 x^{3} h^{2} \cdot - 1 + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.17

Multiply $- 1$ by $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} + 3 x h^{2} (- 2 h x) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.18

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.2.1.4.18.1

Move $x$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} + 3 (x \cdot x) h^{2} (- 2 h) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.18.2

Multiply $x$ by $x$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} + 3 x^{2} h^{2} (- 2 h) + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.19

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

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Step 2.1.2.1.4.19.1

Move $h$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} + 3 x^{2} (h \cdot h^{2}) \cdot - 2 + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.19.2

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

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Step 2.1.2.1.4.19.2.1

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

Step 2.1.2.1.4.19.2.2

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

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} + 3 x^{2} h^{1 + 2} \cdot - 2 + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.19.3

Add $1$ and $2$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} + 3 x^{2} h^{3} \cdot - 2 + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.20

Multiply $- 2$ by $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} + 3 x h^{2} (- h^{2}) + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.21

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

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Step 2.1.2.1.4.21.1

Move $h^{2}$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} + 3 x (h^{2} h^{2}) \cdot - 1 + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.21.2

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

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} + 3 x h^{2 + 2} \cdot - 1 + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.21.3

Add $2$ and $2$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} + 3 x h^{4} \cdot - 1 + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.22

Multiply $- 1$ by $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + h^{3} \cdot 7 + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.23

Move $7$ to the left of $h^{3}$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 \cdot h^{3} + h^{3} (- x^{2}) + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.24

Rewrite using the commutative property of multiplication.

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} + h^{3} (- 2 h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.25

Rewrite using the commutative property of multiplication.

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 h^{3} (h x) + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.26

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

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Step 2.1.2.1.4.26.1

Move $h$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 (h \cdot h^{3}) x + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.26.2

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

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Step 2.1.2.1.4.26.2.1

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

Step 2.1.2.1.4.26.2.2

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

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 h^{1 + 3} x + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.26.3

Add $1$ and $3$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 h^{4} x + h^{3} (- h^{2}) + 98.6$

Step 2.1.2.1.4.27

Rewrite using the commutative property of multiplication.

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 h^{4} x - h^{3} h^{2} + 98.6$

Step 2.1.2.1.4.28

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

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Step 2.1.2.1.4.28.1

Move $h^{2}$ .

$T (x + h) = 7 x^{3} - x^{5} - 2 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 3 x^{4} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 h^{4} x - (h^{2} h^{3}) + 98.6$

Step 2.1.2.1.4.28.2

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

Step 2.1.2.1.4.28.3

Add $2$ and $3$ .

Step 2.1.2.1.5

Subtract $3 x^{4} h$ from $- 2 x^{4} h$ .

$T (x + h) = 7 x^{3} - x^{5} - 5 x^{4} h - x^{3} h^{2} + 21 x^{2} h - 6 x^{3} h^{2} - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 h^{4} x - h^{5} + 98.6$

Step 2.1.2.1.6

Subtract $6 x^{3} h^{2}$ from $- x^{3} h^{2}$ .

$T (x + h) = 7 x^{3} - x^{5} - 5 x^{4} h - 7 x^{3} h^{2} + 21 x^{2} h - 3 x^{2} h^{3} + 21 x h^{2} - 3 x^{3} h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 h^{4} x - h^{5} + 98.6$

Step 2.1.2.1.7

Subtract $3 x^{3} h^{2}$ from $- 7 x^{3} h^{2}$ .

$T (x + h) = 7 x^{3} - x^{5} - 5 x^{4} h - 10 x^{3} h^{2} + 21 x^{2} h - 3 x^{2} h^{3} + 21 x h^{2} - 6 x^{2} h^{3} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 h^{4} x - h^{5} + 98.6$

Step 2.1.2.1.8

Subtract $6 x^{2} h^{3}$ from $- 3 x^{2} h^{3}$ .

$T (x + h) = 7 x^{3} - x^{5} - 5 x^{4} h - 10 x^{3} h^{2} + 21 x^{2} h - 9 x^{2} h^{3} + 21 x h^{2} - 3 x h^{4} + 7 h^{3} - h^{3} x^{2} - 2 h^{4} x - h^{5} + 98.6$

Step 2.1.2.1.9

Subtract $h^{3} x^{2}$ from $- 9 x^{2} h^{3}$ .

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Step 2.1.2.1.9.1

Move $x^{2}$ .

$T (x + h) = 7 x^{3} - x^{5} - 5 x^{4} h - 10 x^{3} h^{2} + 21 x^{2} h + 21 x h^{2} - 3 x h^{4} + 7 h^{3} - 9 h^{3} x^{2} - h^{3} x^{2} - 2 h^{4} x - h^{5} + 98.6$

Step 2.1.2.1.9.2

Subtract $h^{3} x^{2}$ from $- 9 h^{3} x^{2}$ .

$T (x + h) = 7 x^{3} - x^{5} - 5 x^{4} h - 10 x^{3} h^{2} + 21 x^{2} h + 21 x h^{2} - 3 x h^{4} + 7 h^{3} - 10 h^{3} x^{2} - 2 h^{4} x - h^{5} + 98.6$

Step 2.1.2.1.10

Subtract $2 h^{4} x$ from $- 3 x h^{4}$ .

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Step 2.1.2.1.10.1

Move $x$ .

$T (x + h) = 7 x^{3} - x^{5} - 5 x^{4} h - 10 x^{3} h^{2} + 21 x^{2} h + 21 x h^{2} + 7 h^{3} - 10 h^{3} x^{2} - 3 h^{4} x - 2 h^{4} x - h^{5} + 98.6$

Step 2.1.2.1.10.2

Subtract $2 h^{4} x$ from $- 3 h^{4} x$ .

$T (x + h) = 7 x^{3} - x^{5} - 5 x^{4} h - 10 x^{3} h^{2} + 21 x^{2} h + 21 x h^{2} + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 98.6$

Step 2.1.2.2

The final answer is $7 x^{3} - x^{5} - 5 x^{4} h - 10 x^{3} h^{2} + 21 x^{2} h + 21 x h^{2} + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 98.6$ .

$7 x^{3} - x^{5} - 5 x^{4} h - 10 x^{3} h^{2} + 21 x^{2} h + 21 x h^{2} + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 98.6$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{4}$ .

$7 x^{3} - x^{5} - 5 h x^{4} - 10 x^{3} h^{2} + 21 x^{2} h + 21 x h^{2} + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 98.6$

Step 2.2.2

Move $x^{3}$ .

$7 x^{3} - x^{5} - 5 h x^{4} - 10 h^{2} x^{3} + 21 x^{2} h + 21 x h^{2} + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 98.6$

Step 2.2.3

Move $x^{2}$ .

$7 x^{3} - x^{5} - 5 h x^{4} - 10 h^{2} x^{3} + 21 h x^{2} + 21 x h^{2} + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 98.6$

Step 2.2.4

Move $x$ .

$7 x^{3} - x^{5} - 5 h x^{4} - 10 h^{2} x^{3} + 21 h x^{2} + 21 h^{2} x + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 98.6$

Step 2.2.5

Move $7 x^{3}$ .

$- x^{5} - 5 h x^{4} - 10 h^{2} x^{3} + 21 h x^{2} + 21 h^{2} x + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 7 x^{3} + 98.6$

Step 2.2.6

Move $21 h x^{2}$ .

$- x^{5} - 5 h x^{4} - 10 h^{2} x^{3} + 21 h^{2} x + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 21 h x^{2} + 7 x^{3} + 98.6$

Step 2.2.7

Move $21 h^{2} x$ .

$- x^{5} - 5 h x^{4} - 10 h^{2} x^{3} + 7 h^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6$

Step 2.2.8

Move $7 h^{3}$ .

$- x^{5} - 5 h x^{4} - 10 h^{2} x^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6$

Step 2.2.9

Move $- x^{5}$ .

$- 5 h x^{4} - 10 h^{2} x^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6$

Step 2.2.10

Move $- 5 h x^{4}$ .

$- 10 h^{2} x^{3} - 10 h^{3} x^{2} - 5 h^{4} x - h^{5} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6$

Step 2.2.11

Move $- 10 h^{2} x^{3}$ .

$- 10 h^{3} x^{2} - 5 h^{4} x - h^{5} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6$

Step 2.2.12

Move $- 10 h^{3} x^{2}$ .

$- 5 h^{4} x - h^{5} - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6$

Step 2.2.13

Reorder $- 5 h^{4} x$ and $- h^{5}$ .

$- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6$

Step 2.3

Find the components of the definition.

$T (x + h) = - h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6$

$T (x) = 7 x^{3} - x^{5} + 98.6$

$T (x + h) = - h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6$

$T (x) = 7 x^{3} - x^{5} + 98.6$

Step 3

Plug in the components.

$\frac{T (x + h) - T x}{h} = \frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6 - (7 x^{3} - x^{5} + 98.6)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6 - (7 x^{3}) - - x^{5} - 1 \cdot 98.6}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $7$ by $- 1$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6 - 7 x^{3} - - x^{5} - 1 \cdot 98.6}{h}$

Step 4.1.2.2

Multiply $- - x^{5}$ .

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Step 4.1.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6 - 7 x^{3} + 1 x^{5} - 1 \cdot 98.6}{h}$

Step 4.1.2.2.2

Multiply $x^{5}$ by $1$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6 - 7 x^{3} + x^{5} - 1 \cdot 98.6}{h}$

Step 4.1.2.3

Multiply $- 1$ by $98.6$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} - x^{5} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6 - 7 x^{3} + x^{5} - 98.6}{h}$

Step 4.1.3

Add $- x^{5}$ and $x^{5}$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6 - 7 x^{3} + 0 - 98.6}{h}$

Step 4.1.4

Add $- h^{5}$ and $0$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 7 x^{3} + 98.6 - 7 x^{3} - 98.6}{h}$

Step 4.1.5

Subtract $7 x^{3}$ from $7 x^{3}$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 0 + 98.6 - 98.6}{h}$

Step 4.1.6

Add $- h^{5}$ and $0$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 98.6 - 98.6}{h}$

Step 4.1.7

Subtract $98.6$ from $98.6$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2} + 0}{h}$

Step 4.1.8

Add $- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2}$ and $0$ .

$\frac{- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2}}{h}$

Step 4.1.9

Factor $h$ out of $- h^{5} - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2}$ .

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Step 4.1.9.1

Factor $h$ out of $- h^{5}$ .

$\frac{h (- h^{4}) - 5 h^{4} x - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2}}{h}$

Step 4.1.9.2

Factor $h$ out of $- 5 h^{4} x$ .

$\frac{h (- h^{4}) + h (- 5 h^{3} x) - 10 h^{3} x^{2} - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2}}{h}$

Step 4.1.9.3

Factor $h$ out of $- 10 h^{3} x^{2}$ .

$\frac{h (- h^{4}) + h (- 5 h^{3} x) + h (- 10 h^{2} x^{2}) - 10 h^{2} x^{3} - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2}}{h}$

Step 4.1.9.4

Factor $h$ out of $- 10 h^{2} x^{3}$ .

$\frac{h (- h^{4}) + h (- 5 h^{3} x) + h (- 10 h^{2} x^{2}) + h (- 10 h x^{3}) - 5 h x^{4} + 7 h^{3} + 21 h^{2} x + 21 h x^{2}}{h}$

Step 4.1.9.5

Factor $h$ out of $- 5 h x^{4}$ .

$\frac{h (- h^{4}) + h (- 5 h^{3} x) + h (- 10 h^{2} x^{2}) + h (- 10 h x^{3}) + h (- 5 x^{4}) + 7 h^{3} + 21 h^{2} x + 21 h x^{2}}{h}$

Step 4.1.9.6

Factor $h$ out of $7 h^{3}$ .

$\frac{h (- h^{4}) + h (- 5 h^{3} x) + h (- 10 h^{2} x^{2}) + h (- 10 h x^{3}) + h (- 5 x^{4}) + h (7 h^{2}) + 21 h^{2} x + 21 h x^{2}}{h}$

Step 4.1.9.7

Factor $h$ out of $21 h^{2} x$ .

$\frac{h (- h^{4}) + h (- 5 h^{3} x) + h (- 10 h^{2} x^{2}) + h (- 10 h x^{3}) + h (- 5 x^{4}) + h (7 h^{2}) + h (21 h x) + 21 h x^{2}}{h}$

Step 4.1.9.8

Factor $h$ out of $21 h x^{2}$ .

$\frac{h (- h^{4}) + h (- 5 h^{3} x) + h (- 10 h^{2} x^{2}) + h (- 10 h x^{3}) + h (- 5 x^{4}) + h (7 h^{2}) + h (21 h x) + h (21 x^{2})}{h}$

Step 4.1.9.9

Factor $h$ out of $h (- h^{4}) + h (- 5 h^{3} x)$ .

$\frac{h (- h^{4} - 5 h^{3} x) + h (- 10 h^{2} x^{2}) + h (- 10 h x^{3}) + h (- 5 x^{4}) + h (7 h^{2}) + h (21 h x) + h (21 x^{2})}{h}$

Step 4.1.9.10

Factor $h$ out of $h (- h^{4} - 5 h^{3} x) + h (- 10 h^{2} x^{2})$ .

$\frac{h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2}) + h (- 10 h x^{3}) + h (- 5 x^{4}) + h (7 h^{2}) + h (21 h x) + h (21 x^{2})}{h}$

Step 4.1.9.11

Factor $h$ out of $h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2}) + h (- 10 h x^{3})$ .

$\frac{h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3}) + h (- 5 x^{4}) + h (7 h^{2}) + h (21 h x) + h (21 x^{2})}{h}$

Step 4.1.9.12

Factor $h$ out of $h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3}) + h (- 5 x^{4})$ .

$\frac{h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4}) + h (7 h^{2}) + h (21 h x) + h (21 x^{2})}{h}$

Step 4.1.9.13

Factor $h$ out of $h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4}) + h (7 h^{2})$ .

$\frac{h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2}) + h (21 h x) + h (21 x^{2})}{h}$

Step 4.1.9.14

Factor $h$ out of $h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2}) + h (21 h x)$ .

$\frac{h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2} + 21 h x) + h (21 x^{2})}{h}$

Step 4.1.9.15

Factor $h$ out of $h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2} + 21 h x) + h (21 x^{2})$ .

$\frac{h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2} + 21 h x + 21 x^{2})}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{h (- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2} + 21 h x + 21 x^{2})}{h}$

Step 4.2.1.2

Divide $- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2} + 21 h x + 21 x^{2}$ by $1$ .

$- h^{4} - 5 h^{3} x - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2} + 21 h x + 21 x^{2}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h^{3}$ .

$- h^{4} - 5 x h^{3} - 10 h^{2} x^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2} + 21 h x + 21 x^{2}$

Step 4.2.2.2

Move $h^{2}$ .

$- h^{4} - 5 x h^{3} - 10 x^{2} h^{2} - 10 h x^{3} - 5 x^{4} + 7 h^{2} + 21 h x + 21 x^{2}$

Step 4.2.2.3

Move $h$ .

$- h^{4} - 5 x h^{3} - 10 x^{2} h^{2} - 10 x^{3} h - 5 x^{4} + 7 h^{2} + 21 h x + 21 x^{2}$

Step 4.2.2.4

Move $h$ .

$- h^{4} - 5 x h^{3} - 10 x^{2} h^{2} - 10 x^{3} h - 5 x^{4} + 7 h^{2} + 21 x h + 21 x^{2}$

Step 4.2.2.5

Move $7 h^{2}$ .

$- h^{4} - 5 x h^{3} - 10 x^{2} h^{2} - 10 x^{3} h - 5 x^{4} + 21 x h + 21 x^{2} + 7 h^{2}$

Step 4.2.2.6

Move $21 x h$ .

$- h^{4} - 5 x h^{3} - 10 x^{2} h^{2} - 10 x^{3} h - 5 x^{4} + 21 x^{2} + 21 x h + 7 h^{2}$

Step 4.2.2.7

Move $- h^{4}$ .

$- 5 x h^{3} - 10 x^{2} h^{2} - 10 x^{3} h - 5 x^{4} - h^{4} + 21 x^{2} + 21 x h + 7 h^{2}$

Step 4.2.2.8

Move $- 5 x h^{3}$ .

$- 10 x^{2} h^{2} - 10 x^{3} h - 5 x^{4} - 5 x h^{3} - h^{4} + 21 x^{2} + 21 x h + 7 h^{2}$

Step 4.2.2.9

Move $- 10 x^{2} h^{2}$ .

$- 10 x^{3} h - 5 x^{4} - 10 x^{2} h^{2} - 5 x h^{3} - h^{4} + 21 x^{2} + 21 x h + 7 h^{2}$

Step 4.2.2.10

Reorder $- 10 x^{3} h$ and $- 5 x^{4}$ .

$- 5 x^{4} - 10 x^{3} h - 10 x^{2} h^{2} - 5 x h^{3} - h^{4} + 21 x^{2} + 21 x h + 7 h^{2}$

Step 5