Precalculus Examples

$\frac{25 - x^{2}}{x^{3} - 125}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite $25$ as $5^{2}$ .

$\frac{5^{2} - x^{2}}{x^{3} - 125}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$\frac{(5 + x) (5 - x)}{x^{3} - 125}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

Rewrite $125$ as $5^{3}$ .

$\frac{(5 + x) (5 - x)}{x^{3} - 5^{3}}$

Step 2.2

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 $a = x$ and $b = 5$ .

$\frac{(5 + x) (5 - x)}{(x - 5) (x^{2} + x \cdot 5 + 5^{2})}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Move $5$ to the left of $x$ .

$\frac{(5 + x) (5 - x)}{(x - 5) (x^{2} + 5 \cdot x + 5^{2})}$

Step 2.3.2

Raise $5$ to the power of $2$ .

$\frac{(5 + x) (5 - x)}{(x - 5) (x^{2} + 5 x + 25)}$

Step 3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 3.1

Cancel the common factor of $5 - x$ and $x - 5$ .

Tap for more steps...

Step 3.1.1

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{(5 + x) (- 1 (- 5) - x)}{(x - 5) (x^{2} + 5 x + 25)}$

Step 3.1.2

Factor $- 1$ out of $- x$ .

$\frac{(5 + x) (- 1 (- 5) - (x))}{(x - 5) (x^{2} + 5 x + 25)}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- 5) - (x)$ .

$\frac{(5 + x) (- 1 (- 5 + x))}{(x - 5) (x^{2} + 5 x + 25)}$

Step 3.1.4

Reorder terms.

$\frac{(5 + x) (- 1 (x - 5))}{(x - 5) (x^{2} + 5 x + 25)}$

Step 3.1.5

Cancel the common factor.

$\frac{(5 + x) (- 1 (x - 5))}{(x - 5) (x^{2} + 5 x + 25)}$

Step 3.1.6

Rewrite the expression.

$\frac{(5 + x) \cdot (- 1)}{x^{2} + 5 x + 25}$

Step 3.2

Simplify the expression.

Tap for more steps...

Step 3.2.1

Move $- 1$ to the left of $5 + x$ .

$\frac{- 1 \cdot (5 + x)}{x^{2} + 5 x + 25}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{5 + x}{x^{2} + 5 x + 25}$