Precalculus Examples

$- x^{3} + 24 x^{2} - 179 x + 495$

Step 1

Write $- x^{3} + 24 x^{2} - 179 x + 495$ as a function.

$f (x) = - x^{3} + 24 x^{2} - 179 x + 495$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (x + h) = - {(x + h)}^{3} + 24 {(x + h)}^{2} - 179 (x + h) + 495$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Use the Binomial Theorem.

$f (x + h) = - (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) + 24 {(x + h)}^{2} - 179 (x + h) + 495$

Step 3.1.2.1.2

Apply the distributive property.

$f (x + h) = - x^{3} - (3 x^{2} h) - (3 x h^{2}) - h^{3} + 24 {(x + h)}^{2} - 179 (x + h) + 495$

Step 3.1.2.1.3

Simplify.

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Step 3.1.2.1.3.1

Multiply $3$ by $- 1$ .

$f (x + h) = - x^{3} - 3 (x^{2} h) - (3 x h^{2}) - h^{3} + 24 {(x + h)}^{2} - 179 (x + h) + 495$

Step 3.1.2.1.3.2

Multiply $3$ by $- 1$ .

$f (x + h) = - x^{3} - 3 (x^{2} h) - 3 (x h^{2}) - h^{3} + 24 {(x + h)}^{2} - 179 (x + h) + 495$

Step 3.1.2.1.4

Remove parentheses.

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 {(x + h)}^{2} - 179 (x + h) + 495$

Step 3.1.2.1.5

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

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 ((x + h) (x + h)) - 179 (x + h) + 495$

Step 3.1.2.1.6

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

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Step 3.1.2.1.6.1

Apply the distributive property.

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 (x (x + h) + h (x + h)) - 179 (x + h) + 495$

Step 3.1.2.1.6.2

Apply the distributive property.

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 (x \cdot x + x h + h (x + h)) - 179 (x + h) + 495$

Step 3.1.2.1.6.3

Apply the distributive property.

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 (x \cdot x + x h + h x + h \cdot h) - 179 (x + h) + 495$

Step 3.1.2.1.7

Simplify and combine like terms.

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Step 3.1.2.1.7.1

Simplify each term.

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Step 3.1.2.1.7.1.1

Multiply $x$ by $x$ .

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 (x^{2} + x h + h x + h \cdot h) - 179 (x + h) + 495$

Step 3.1.2.1.7.1.2

Multiply $h$ by $h$ .

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 (x^{2} + x h + h x + h^{2}) - 179 (x + h) + 495$

Step 3.1.2.1.7.2

Add $x h$ and $h x$ .

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Step 3.1.2.1.7.2.1

Reorder $x$ and $h$ .

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 (x^{2} + h x + h x + h^{2}) - 179 (x + h) + 495$

Step 3.1.2.1.7.2.2

Add $h x$ and $h x$ .

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 (x^{2} + 2 h x + h^{2}) - 179 (x + h) + 495$

Step 3.1.2.1.8

Apply the distributive property.

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 x^{2} + 24 (2 h x) + 24 h^{2} - 179 (x + h) + 495$

Step 3.1.2.1.9

Multiply $2$ by $24$ .

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 x^{2} + 48 (h x) + 24 h^{2} - 179 (x + h) + 495$

Step 3.1.2.1.10

Apply the distributive property.

$f (x + h) = - x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 x^{2} + 48 h x + 24 h^{2} - 179 x - 179 h + 495$

Step 3.1.2.2

The final answer is $- x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 x^{2} + 48 h x + 24 h^{2} - 179 x - 179 h + 495$ .

$- x^{3} - 3 x^{2} h - 3 x h^{2} - h^{3} + 24 x^{2} + 48 h x + 24 h^{2} - 179 x - 179 h + 495$

Step 3.2

Reorder.

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Step 3.2.1

Move $x^{2}$ .

$- x^{3} - 3 h x^{2} - 3 x h^{2} - h^{3} + 24 x^{2} + 48 h x + 24 h^{2} - 179 x - 179 h + 495$

Step 3.2.2

Move $x$ .

$- x^{3} - 3 h x^{2} - 3 h^{2} x - h^{3} + 24 x^{2} + 48 h x + 24 h^{2} - 179 x - 179 h + 495$

Step 3.2.3

Move $- 179 x$ .

$- x^{3} - 3 h x^{2} - 3 h^{2} x - h^{3} + 24 x^{2} + 48 h x + 24 h^{2} - 179 h - 179 x + 495$

Step 3.2.4

Move $24 x^{2}$ .

$- x^{3} - 3 h x^{2} - 3 h^{2} x - h^{3} + 48 h x + 24 h^{2} + 24 x^{2} - 179 h - 179 x + 495$

Step 3.2.5

Move $48 h x$ .

$- x^{3} - 3 h x^{2} - 3 h^{2} x - h^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495$

Step 3.2.6

Move $- x^{3}$ .

$- 3 h x^{2} - 3 h^{2} x - h^{3} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495$

Step 3.2.7

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

$- 3 h^{2} x - h^{3} - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495$

Step 3.2.8

Reorder $- 3 h^{2} x$ and $- h^{3}$ .

$- h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495$

Step 3.3

Find the components of the definition.

$f (x + h) = - h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495$

$f (x) = - x^{3} + 24 x^{2} - 179 x + 495$

$f (x + h) = - h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495$

$f (x) = - x^{3} + 24 x^{2} - 179 x + 495$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495 - (- x^{3} + 24 x^{2} - 179 x + 495)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495 - - x^{3} - (24 x^{2}) - (- 179 x) - 1 \cdot 495}{h}$

Step 5.1.2

Simplify.

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Step 5.1.2.1

Multiply $- - x^{3}$ .

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Step 5.1.2.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495 + 1 x^{3} - (24 x^{2}) - (- 179 x) - 1 \cdot 495}{h}$

Step 5.1.2.1.2

Multiply $x^{3}$ by $1$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495 + x^{3} - (24 x^{2}) - (- 179 x) - 1 \cdot 495}{h}$

Step 5.1.2.2

Multiply $24$ by $- 1$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495 + x^{3} - 24 x^{2} - (- 179 x) - 1 \cdot 495}{h}$

Step 5.1.2.3

Multiply $- 179$ by $- 1$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495 + x^{3} - 24 x^{2} + 179 x - 1 \cdot 495}{h}$

Step 5.1.2.4

Multiply $- 1$ by $495$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} - x^{3} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495 + x^{3} - 24 x^{2} + 179 x - 495}{h}$

Step 5.1.3

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

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495 + 0 - 24 x^{2} + 179 x - 495}{h}$

Step 5.1.4

Add $- h^{3}$ and $0$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x + 24 x^{2} - 179 h - 179 x + 495 - 24 x^{2} + 179 x - 495}{h}$

Step 5.1.5

Subtract $24 x^{2}$ from $24 x^{2}$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h - 179 x + 495 + 0 + 179 x - 495}{h}$

Step 5.1.6

Add $- h^{3}$ and $0$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h - 179 x + 495 + 179 x - 495}{h}$

Step 5.1.7

Add $- 179 x$ and $179 x$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h + 0 + 495 - 495}{h}$

Step 5.1.8

Add $- h^{3}$ and $0$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h + 495 - 495}{h}$

Step 5.1.9

Subtract $495$ from $495$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h + 0}{h}$

Step 5.1.10

Add $- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h$ and $0$ .

$\frac{- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h}{h}$

Step 5.1.11

Factor $h$ out of $- h^{3} - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h$ .

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Step 5.1.11.1

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

$\frac{h (- h^{2}) - 3 h^{2} x - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h}{h}$

Step 5.1.11.2

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

$\frac{h (- h^{2}) + h (- 3 h x) - 3 h x^{2} + 24 h^{2} + 48 h x - 179 h}{h}$

Step 5.1.11.3

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

$\frac{h (- h^{2}) + h (- 3 h x) + h (- 3 x^{2}) + 24 h^{2} + 48 h x - 179 h}{h}$

Step 5.1.11.4

Factor $h$ out of $24 h^{2}$ .

$\frac{h (- h^{2}) + h (- 3 h x) + h (- 3 x^{2}) + h (24 h) + 48 h x - 179 h}{h}$

Step 5.1.11.5

Factor $h$ out of $48 h x$ .

$\frac{h (- h^{2}) + h (- 3 h x) + h (- 3 x^{2}) + h (24 h) + h (48 x) - 179 h}{h}$

Step 5.1.11.6

Factor $h$ out of $- 179 h$ .

$\frac{h (- h^{2}) + h (- 3 h x) + h (- 3 x^{2}) + h (24 h) + h (48 x) + h \cdot - 179}{h}$

Step 5.1.11.7

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

$\frac{h (- h^{2} - 3 h x) + h (- 3 x^{2}) + h (24 h) + h (48 x) + h \cdot - 179}{h}$

Step 5.1.11.8

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

$\frac{h (- h^{2} - 3 h x - 3 x^{2}) + h (24 h) + h (48 x) + h \cdot - 179}{h}$

Step 5.1.11.9

Factor $h$ out of $h (- h^{2} - 3 h x - 3 x^{2}) + h (24 h)$ .

$\frac{h (- h^{2} - 3 h x - 3 x^{2} + 24 h) + h (48 x) + h \cdot - 179}{h}$

Step 5.1.11.10

Factor $h$ out of $h (- h^{2} - 3 h x - 3 x^{2} + 24 h) + h (48 x)$ .

$\frac{h (- h^{2} - 3 h x - 3 x^{2} + 24 h + 48 x) + h \cdot - 179}{h}$

Step 5.1.11.11

Factor $h$ out of $h (- h^{2} - 3 h x - 3 x^{2} + 24 h + 48 x) + h \cdot - 179$ .

$\frac{h (- h^{2} - 3 h x - 3 x^{2} + 24 h + 48 x - 179)}{h}$

Step 5.2

Reduce the expression by cancelling the common factors.

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Step 5.2.1

Cancel the common factor of $h$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{h (- h^{2} - 3 h x - 3 x^{2} + 24 h + 48 x - 179)}{h}$

Step 5.2.1.2

Divide $- h^{2} - 3 h x - 3 x^{2} + 24 h + 48 x - 179$ by $1$ .

$- h^{2} - 3 h x - 3 x^{2} + 24 h + 48 x - 179$

Step 5.2.2

Simplify the expression.

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Step 5.2.2.1

Move $h$ .

$- h^{2} - 3 x h - 3 x^{2} + 24 h + 48 x - 179$

Step 5.2.2.2

Move $24 h$ .

$- h^{2} - 3 x h - 3 x^{2} + 48 x + 24 h - 179$

Step 5.2.2.3

Move $- h^{2}$ .

$- 3 x h - 3 x^{2} - h^{2} + 48 x + 24 h - 179$

Step 5.2.2.4

Reorder $- 3 x h$ and $- 3 x^{2}$ .

$- 3 x^{2} - 3 x h - h^{2} + 48 x + 24 h - 179$

Step 6