Algebra Examples

$1 - \frac{1}{8} p^{3}$

Step 1

Rewrite $1$ as $1^{3}$ .

$1^{3} - \frac{1}{8} p^{3}$

Step 2

Rewrite $\frac{1}{8} p^{3}$ as ${(\frac{1}{2} p)}^{3}$ .

$1^{3} - {(\frac{1}{2} p)}^{3}$

Step 3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = \frac{1}{2} p$ .

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

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{2}$ and $p$ .

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

Step 4.2

One to any power is one.

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

Step 4.3

Multiply $\frac{1}{2} p$ by $1$ .

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

Step 4.4

Combine $\frac{1}{2}$ and $p$ .

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

Step 4.5

Combine $\frac{1}{2}$ and $p$ .

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

Step 4.6

Apply the product rule to $\frac{p}{2}$ .

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

Step 4.7

Raise $2$ to the power of $2$ .

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