Precalculus Examples

$\frac{{(x + h)}^{3} - 7 (x + h) - (x^{3} - 7 x)}{h}$

Step 1

Simplify the numerator.

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Step 1.1

Use the Binomial Theorem.

$\frac{x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 7 (x + h) - (x^{3} - 7 x)}{h}$

Step 1.2

Apply the distributive property.

$\frac{x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 7 x - 7 h - (x^{3} - 7 x)}{h}$

Step 1.3

Apply the distributive property.

$\frac{x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 7 x - 7 h - x^{3} - (- 7 x)}{h}$

Step 1.4

Multiply $- 7$ by $- 1$ .

$\frac{x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 7 x - 7 h - x^{3} + 7 x}{h}$

Step 1.5

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

$\frac{3 x^{2} h + 3 x h^{2} + h^{3} - 7 x - 7 h + 0 + 7 x}{h}$

Step 1.6

Add $3 x^{2} h$ and $0$ .

$\frac{3 x^{2} h + 3 x h^{2} + h^{3} - 7 x - 7 h + 7 x}{h}$

Step 1.7

Add $- 7 x$ and $7 x$ .

$\frac{3 x^{2} h + 3 x h^{2} + h^{3} - 7 h + 0}{h}$

Step 1.8

Add $3 x^{2} h + 3 x h^{2} + h^{3} - 7 h$ and $0$ .

$\frac{3 x^{2} h + 3 x h^{2} + h^{3} - 7 h}{h}$

Step 1.9

Factor $h$ out of $3 x^{2} h + 3 x h^{2} + h^{3} - 7 h$ .

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Step 1.9.1

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

$\frac{h (3 x^{2}) + 3 x h^{2} + h^{3} - 7 h}{h}$

Step 1.9.2

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

$\frac{h (3 x^{2}) + h (3 x h) + h^{3} - 7 h}{h}$

Step 1.9.3

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

$\frac{h (3 x^{2}) + h (3 x h) + h \cdot h^{2} - 7 h}{h}$

Step 1.9.4

Factor $h$ out of $- 7 h$ .

$\frac{h (3 x^{2}) + h (3 x h) + h \cdot h^{2} + h \cdot - 7}{h}$

Step 1.9.5

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

$\frac{h (3 x^{2} + 3 x h) + h \cdot h^{2} + h \cdot - 7}{h}$

Step 1.9.6

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

$\frac{h (3 x^{2} + 3 x h + h^{2}) + h \cdot - 7}{h}$

Step 1.9.7

Factor $h$ out of $h (3 x^{2} + 3 x h + h^{2}) + h \cdot - 7$ .

$\frac{h (3 x^{2} + 3 x h + h^{2} - 7)}{h}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $h$ .

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Step 2.1.1

Cancel the common factor.

$\frac{h (3 x^{2} + 3 x h + h^{2} - 7)}{h}$

Step 2.1.2

Divide $3 x^{2} + 3 x h + h^{2} - 7$ by $1$ .

$3 x^{2} + 3 x h + h^{2} - 7$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $x$ .

$3 x^{2} + 3 h x + h^{2} - 7$

Step 2.2.2

Move $3 x^{2}$ .

$3 h x + h^{2} + 3 x^{2} - 7$

Step 2.2.3

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

$h^{2} + 3 h x + 3 x^{2} - 7$