Precalculus Examples

$\frac{{(x + h)}^{4} - x^{4}}{h}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite ${(x + h)}^{4}$ as ${({(x + h)}^{2})}^{2}$ .

$\frac{{({(x + h)}^{2})}^{2} - x^{4}}{h}$

Step 1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

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

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Step 1.4.2.1

Apply the distributive property.

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

Step 1.4.2.2

Apply the distributive property.

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

Step 1.4.2.3

Apply the distributive property.

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

Step 1.4.3

Simplify and combine like terms.

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Multiply $x$ by $x$ .

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

Step 1.4.3.1.2

Multiply $h$ by $h$ .

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

Step 1.4.3.2

Add $x h$ and $h x$ .

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Step 1.4.3.2.1

Reorder $x$ and $h$ .

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

Step 1.4.3.2.2

Add $h x$ and $h x$ .

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

Step 1.4.4

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

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

Step 1.4.5

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

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

Step 1.4.6

Simplify.

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Step 1.4.6.1

Add $x$ and $x$ .

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

Step 1.4.6.2

Subtract $x$ from $x$ .

$\frac{(2 x^{2} + 2 h x + h^{2}) ((h + 2 x) (h + 0))}{h}$

Step 1.4.6.3

Add $h$ and $0$ .

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

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

Step 2

Cancel the common factor of $h$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $(2 x^{2} + 2 h x + h^{2}) (h + 2 x)$ by $1$ .

$(2 x^{2} + 2 h x + h^{2}) (h + 2 x)$

Step 3

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

$2 x^{2} h + 2 x^{2} (2 x) + 2 h x h + 2 h x (2 x) + h^{2} h + h^{2} (2 x)$

Step 4

Simplify each term.

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Step 4.1

Rewrite using the commutative property of multiplication.

$2 x^{2} h + 2 \cdot 2 x^{2} x + 2 h x h + 2 h x (2 x) + h^{2} h + h^{2} (2 x)$

Step 4.2

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

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Step 4.2.1

Move $x$ .

$2 x^{2} h + 2 \cdot 2 (x \cdot x^{2}) + 2 h x h + 2 h x (2 x) + h^{2} h + h^{2} (2 x)$

Step 4.2.2

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

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Step 4.2.2.1

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

$2 x^{2} h + 2 \cdot 2 (x^{1} x^{2}) + 2 h x h + 2 h x (2 x) + h^{2} h + h^{2} (2 x)$

Step 4.2.2.2

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

$2 x^{2} h + 2 \cdot 2 x^{1 + 2} + 2 h x h + 2 h x (2 x) + h^{2} h + h^{2} (2 x)$

Step 4.2.3

Add $1$ and $2$ .

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

Step 4.3

Multiply $2$ by $2$ .

$2 x^{2} h + 4 x^{3} + 2 h x h + 2 h x (2 x) + h^{2} h + h^{2} (2 x)$

Step 4.4

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

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Step 4.4.1

Move $h$ .

$2 x^{2} h + 4 x^{3} + 2 (h \cdot h) x + 2 h x (2 x) + h^{2} h + h^{2} (2 x)$

Step 4.4.2

Multiply $h$ by $h$ .

$2 x^{2} h + 4 x^{3} + 2 h^{2} x + 2 h x (2 x) + h^{2} h + h^{2} (2 x)$

Step 4.5

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

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Step 4.5.1

Move $x$ .

$2 x^{2} h + 4 x^{3} + 2 h^{2} x + 2 h (x \cdot x) \cdot 2 + h^{2} h + h^{2} (2 x)$

Step 4.5.2

Multiply $x$ by $x$ .

$2 x^{2} h + 4 x^{3} + 2 h^{2} x + 2 h x^{2} \cdot 2 + h^{2} h + h^{2} (2 x)$

Step 4.6

Multiply $2$ by $2$ .

$2 x^{2} h + 4 x^{3} + 2 h^{2} x + 4 h x^{2} + h^{2} h + h^{2} (2 x)$

Step 4.7

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

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Step 4.7.1

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

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Step 4.7.1.1

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

$2 x^{2} h + 4 x^{3} + 2 h^{2} x + 4 h x^{2} + h^{2} h^{1} + h^{2} (2 x)$

Step 4.7.1.2

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

$2 x^{2} h + 4 x^{3} + 2 h^{2} x + 4 h x^{2} + h^{2 + 1} + h^{2} (2 x)$

Step 4.7.2

Add $2$ and $1$ .

$2 x^{2} h + 4 x^{3} + 2 h^{2} x + 4 h x^{2} + h^{3} + h^{2} (2 x)$

Step 4.8

Rewrite using the commutative property of multiplication.

$2 x^{2} h + 4 x^{3} + 2 h^{2} x + 4 h x^{2} + h^{3} + 2 h^{2} x$

Step 5

Add $2 x^{2} h$ and $4 h x^{2}$ .

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Step 5.1

Move $x^{2}$ .

$4 x^{3} + 2 h^{2} x + 2 h x^{2} + 4 h x^{2} + h^{3} + 2 h^{2} x$

Step 5.2

Add $2 h x^{2}$ and $4 h x^{2}$ .

$4 x^{3} + 2 h^{2} x + 6 h x^{2} + h^{3} + 2 h^{2} x$

Step 6

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

$4 x^{3} + 6 h x^{2} + h^{3} + 4 h^{2} x$

Step 7

Reorder.

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Step 7.1

Move $4 x^{3}$ .

$6 h x^{2} + h^{3} + 4 h^{2} x + 4 x^{3}$

Step 7.2

Move $6 h x^{2}$ .

$h^{3} + 4 h^{2} x + 6 h x^{2} + 4 x^{3}$