Algebra Examples

$f (x) = \frac{3 x^{2} + 6 x}{x^{2} + x}$

Step 1

Factor $3 x$ out of $3 x^{2} + 6 x$ .

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Step 1.1

Factor $3 x$ out of $3 x^{2}$ .

$f (x) = \frac{3 x (x) + 6 x}{x^{2} + x}$

Step 1.2

Factor $3 x$ out of $6 x$ .

$f (x) = \frac{3 x (x) + 3 x (2)}{x^{2} + x}$

Step 1.3

Factor $3 x$ out of $3 x (x) + 3 x (2)$ .

$f (x) = \frac{3 x (x + 2)}{x^{2} + x}$

Step 2

Factor $x$ out of $x^{2} + x$ .

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Step 2.1

Factor $x$ out of $x^{2}$ .

$f (x) = \frac{3 x (x + 2)}{x \cdot x + x}$

Step 2.2

Raise $x$ to the power of $1$ .

$f (x) = \frac{3 x (x + 2)}{x \cdot x + x}$

Step 2.3

Factor $x$ out of $x^{1}$ .

$f (x) = \frac{3 x (x + 2)}{x \cdot x + x \cdot 1}$

Step 2.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$f (x) = \frac{3 x (x + 2)}{x (x + 1)}$

Step 3

Cancel the common factor of $x$ .

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Step 3.1

Cancel the common factor.

$f (x) = \frac{3 x (x + 2)}{x (x + 1)}$

Step 3.2

Rewrite the expression.

$f (x) = \frac{3 (x + 2)}{x + 1}$

Step 4

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

$x$

Step 5

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{3 (x + 2)}{x + 1}$ .

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Step 5.1

Set $x$ equal to $0$ .

$x = 0$

Step 5.2

Substitute $0$ for $x$ in $\frac{3 (x + 2)}{x + 1}$ and simplify.

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Step 5.2.1

Substitute $0$ for $x$ to find the $y$ coordinate of the hole.

$\frac{3 (0 + 2)}{0 + 1}$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Add $0$ and $2$ .

$\frac{3 \cdot 2}{0 + 1}$

Step 5.2.2.2

Add $0$ and $1$ .

$\frac{3 \cdot 2}{1}$

Step 5.2.2.3

Multiply $3$ by $2$ .

$\frac{6}{1}$

Step 5.2.2.4

Divide $6$ by $1$ .

$6$

Step 5.3

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

$(0, 6)$

Step 6