Algebra Examples

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

Step 1

Factor $x^{2} + x - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 3)}{x - 2}$

Step 2

Cancel the common factor of $x - 2$ .

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Step 2.1

Cancel the common factor.

$\frac{(x - 2) (x + 3)}{x - 2}$

Step 2.2

Divide $x + 3$ by $1$ .

$x + 3$

Step 3

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

$x - 2$

Step 4

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $x + 3$ .

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Step 4.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.3

Substitute $2$ for $x$ in $x + 3$ and simplify.

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Step 4.3.1

Substitute $2$ for $x$ to find the $y$ coordinate of the hole.

$2 + 3$

Step 4.3.2

Add $2$ and $3$ .

$5$

Step 4.4

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

$(2, 5)$

Step 5