Pre-Algebra Examples

$\frac{16 x^{4} - 1}{2 x - 1}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $16 x^{4}$ as ${(4 x^{2})}^{2}$ .

$\frac{{(4 x^{2})}^{2} - 1}{2 x - 1}$

Step 1.2

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

$\frac{{(4 x^{2})}^{2} - 1^{2}}{2 x - 1}$

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 $a = 4 x^{2}$ and $b = 1$ .

$\frac{(4 x^{2} + 1) (4 x^{2} - 1)}{2 x - 1}$

Step 1.4

Simplify.

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Step 1.4.1

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

$\frac{(4 x^{2} + 1) ({(2 x)}^{2} - 1)}{2 x - 1}$

Step 1.4.2

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

$\frac{(4 x^{2} + 1) ({(2 x)}^{2} - 1^{2})}{2 x - 1}$

Step 1.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

$\frac{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}{2 x - 1}$

Step 2

Cancel the common factor of $2 x - 1$ .

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Step 2.1

Cancel the common factor.

$\frac{(4 x^{2} + 1) (2 x + 1) (2 x - 1)}{2 x - 1}$

Step 2.2

Divide $(4 x^{2} + 1) (2 x + 1)$ by $1$ .

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

Step 3

Expand $(4 x^{2} + 1) (2 x + 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

$4 x^{2} (2 x) + 4 x^{2} \cdot 1 + 1 (2 x + 1)$

Step 3.3

Apply the distributive property.

$4 x^{2} (2 x) + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1$

Step 4

Simplify each term.

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Step 4.1

Rewrite using the commutative property of multiplication.

$4 \cdot 2 x^{2} x + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1$

Step 4.2

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

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Step 4.2.1

Move $x$ .

$4 \cdot 2 (x \cdot x^{2}) + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1$

Step 4.2.2

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

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Step 4.2.2.1

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

$4 \cdot 2 (x^{1} x^{2}) + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1$

Step 4.2.2.2

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

$4 \cdot 2 x^{1 + 2} + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1$

Step 4.2.3

Add $1$ and $2$ .

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

Step 4.3

Multiply $4$ by $2$ .

$8 x^{3} + 4 x^{2} \cdot 1 + 1 (2 x) + 1 \cdot 1$

Step 4.4

Multiply $4$ by $1$ .

$8 x^{3} + 4 x^{2} + 1 (2 x) + 1 \cdot 1$

Step 4.5

Multiply $2 x$ by $1$ .

$8 x^{3} + 4 x^{2} + 2 x + 1 \cdot 1$

Step 4.6

Multiply $1$ by $1$ .

$8 x^{3} + 4 x^{2} + 2 x + 1$