Algebra Examples

$f (x) = x^{2} + 2 x + 1$ ,

$g (x) = x$

Step 1

Find the quotient of the functions.

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Step 1.1

Replace the function designators with the actual functions in

$\frac{f (x)}{g (x)}$ .

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

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite

$1$ as

$1^{2}$ .

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

Step 1.2.2

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

$2 x = 2 \cdot x \cdot 1$

Step 1.2.3

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot 1 + 1^{2}}{x}$

Step 1.2.4

Factor using the perfect square trinomial rule

$a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where

$a = x$ and

$b = 1$ .

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

Step 2

Set the denominator in

$\frac{{(x + 1)}^{2}}{x}$ equal to

$0$ to find where the expression is undefined.

$x = 0$

Step 3

The domain is all values of

$x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4

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