Algebra Examples

$\frac{x + 7}{x^{2} + x - 42}$

Step 1

Factor $x^{2} + x - 42$ 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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Step 1.2

Write the factored form using these integers.

$\frac{x + 7}{(x - 6) (x + 7)}$

Step 2

Cancel the common factor of $x + 7$ .

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Step 2.1

Cancel the common factor.

$\frac{x + 7}{(x - 6) (x + 7)}$

Step 2.2

Rewrite the expression.

$\frac{1}{x - 6}$

Step 3

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

$x + 7$

Step 4

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{1}{x - 6}$ .

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Step 4.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 4.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 4.3

Substitute $- 7$ for $x$ in $\frac{1}{x - 6}$ and simplify.

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Step 4.3.1

Substitute $- 7$ for $x$ to find the $y$ coordinate of the hole.

$\frac{1}{- 7 - 6}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Subtract $6$ from $- 7$ .

$\frac{1}{- 13}$

Step 4.3.2.2

Move the negative in front of the fraction.

$- \frac{1}{13}$

Step 4.4

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

$(- 7, - \frac{1}{13})$

Step 5