Algebra Examples

$\frac{x^{2} + 5 x + 4}{x^{2} - 1}$

Step 1

Factor $x^{2} + 5 x + 4$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $5$ .

$1, 4$

Step 1.2

Write the factored form using these integers.

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

Step 2

Factor $x^{2} - 1$ .

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Step 2.1

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

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

Step 2.2

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 $b = 1$ .

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

Step 3

Cancel the common factor of $x + 1$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

To find the holes in the graph, look at the denominator factors that were cancelled.

$x + 1$

Step 5

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{x + 4}{x - 1}$ .

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Step 5.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5.3

Substitute $- 1$ for $x$ in $\frac{x + 4}{x - 1}$ and simplify.

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Step 5.3.1

Substitute $- 1$ for $x$ to find the $y$ coordinate of the hole.

$\frac{- 1 + 4}{- 1 - 1}$

Step 5.3.2

Simplify.

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Step 5.3.2.1

Add $- 1$ and $4$ .

$\frac{3}{- 1 - 1}$

Step 5.3.2.2

Subtract $1$ from $- 1$ .

$\frac{3}{- 2}$

Step 5.3.2.3

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 5.4

The holes in the graph are the points where any of the cancelled factors are equal to $0$ .

$(- 1, - \frac{3}{2})$

Step 6