Algebra Examples

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

Step 1

Factor using the perfect square rule.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot (2 x) \cdot 1$

Step 1.4

Rewrite the polynomial.

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

Step 1.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 x$ and $b = 1$ .

$\frac{{(2 x - 1)}^{2}}{8 x^{3} - 1}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the product rule to $2 x$ .

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

Step 2.4.2

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

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

Step 2.4.3

Multiply $2$ by $1$ .

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

Step 2.4.4

One to any power is one.

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

Step 3

Cancel the common factor of ${(2 x - 1)}^{2}$ and $2 x - 1$ .

Tap for more steps...

Step 3.1

Factor $2 x - 1$ out of ${(2 x - 1)}^{2}$ .

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

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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