Precalculus Examples

$8 t^{3} - 1$

Step 1

Rewrite $8 t^{3}$ as ${(2 t)}^{3}$ .

${(2 t)}^{3} - 1$

Step 2

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

${(2 t)}^{3} - 1^{3}$

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 $a = 2 t$ and $b = 1$ .

$(2 t - 1) ({(2 t)}^{2} + 2 t \cdot 1 + 1^{2})$

Step 4

Simplify.

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Step 4.1

Apply the product rule to $2 t$ .

$(2 t - 1) (2^{2} t^{2} + 2 t \cdot 1 + 1^{2})$

Step 4.2

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

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

Step 4.3

Multiply $2$ by $1$ .

$(2 t - 1) (4 t^{2} + 2 t + 1^{2})$

Step 4.4

One to any power is one.

$(2 t - 1) (4 t^{2} + 2 t + 1)$