Algebra Examples

$f (x) = \frac{x^{2} - 36}{x + 6}$ , $g (x) = x - 6$

Step 1

Find the product of the functions.

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Step 1.1

Replace the function designators with the actual functions in $f (x) \cdot (g (x))$ .

$(\frac{x^{2} - 36}{x + 6}) \cdot (x - 6)$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Rewrite $36$ as $6^{2}$ .

$\frac{x^{2} - 6^{2}}{x + 6} \cdot (x - 6)$

Step 1.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$\frac{(x + 6) (x - 6)}{x + 6} \cdot (x - 6)$

Step 1.2.2

Cancel the common factor of $x + 6$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{(x + 6) (x - 6)}{x + 6} \cdot (x - 6)$

Step 1.2.2.2

Divide $x - 6$ by $1$ .

$(x - 6) \cdot (x - 6)$

Step 1.2.3

Expand $(x - 6) (x - 6)$ using the FOIL Method.

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Step 1.2.3.1

Apply the distributive property.

$x (x - 6) - 6 (x - 6)$

Step 1.2.3.2

Apply the distributive property.

$x \cdot x + x \cdot - 6 - 6 (x - 6)$

Step 1.2.3.3

Apply the distributive property.

$x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6$

Step 1.2.4

Simplify and combine like terms.

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Step 1.2.4.1

Simplify each term.

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Step 1.2.4.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6$

Step 1.2.4.1.2

Move $- 6$ to the left of $x$ .

$x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6$

Step 1.2.4.1.3

Multiply $- 6$ by $- 6$ .

$x^{2} - 6 x - 6 x + 36$

Step 1.2.4.2

Subtract $6 x$ from $- 6 x$ .

$x^{2} - 12 x + 36$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3