Algebra Examples

$f (x) = \frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{3} - 9 x = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor the left side of the equation.

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Step 1.2.2.1.1

Factor $x$ out of $x^{3} - 9 x$ .

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Step 1.2.2.1.1.1

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

$x \cdot x^{2} - 9 x = 0$

Step 1.2.2.1.1.2

Factor $x$ out of $- 9 x$ .

$x \cdot x^{2} + x \cdot - 9 = 0$

Step 1.2.2.1.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

$x (x^{2} - 9) = 0$

Step 1.2.2.1.2

Rewrite $9$ as $3^{2}$ .

$x (x^{2} - 3^{2}) = 0$

Step 1.2.2.1.3

Factor.

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Step 1.2.2.1.3.1

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 $b = 3$ .

$x ((x + 3) (x - 3)) = 0$

Step 1.2.2.1.3.2

Remove unnecessary parentheses.

$x (x + 3) (x - 3) = 0$

Step 1.2.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 1.2.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.2.4

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.2.4.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 1.2.2.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.2.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.2.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.2.6

The final solution is all the values that make $x (x + 3) (x - 3) = 0$ true.

$x = 0, - 3, 3$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9}$ true.

$x = 0, - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{{(0)}^{3} - 9 \cdot 0}{3 {(0)}^{2} - 6 \cdot 0 - 9}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0^{3} - 9 \cdot 0}{3 {(0)}^{2} - 6 \cdot 0 - 9}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{3} - 9 \cdot 0}{3 \cdot 0^{2} - 6 \cdot 0 - 9}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{3} - 9 \cdot 0}{3 {(0)}^{2} - 6 \cdot 0 - 9}$

Step 2.2.4

Simplify $\frac{{(0)}^{3} - 9 \cdot 0}{3 {(0)}^{2} - 6 \cdot 0 - 9}$ .

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0 - 9 \cdot 0}{3 \cdot 0^{2} - 6 \cdot 0 - 9}$

Step 2.2.4.1.2

Multiply $- 9$ by $0$ .

$y = \frac{0 + 0}{3 \cdot 0^{2} - 6 \cdot 0 - 9}$

Step 2.2.4.1.3

Add $0$ and $0$ .

$y = \frac{0}{3 \cdot 0^{2} - 6 \cdot 0 - 9}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0}{3 \cdot 0 - 6 \cdot 0 - 9}$

Step 2.2.4.2.2

Multiply $3$ by $0$ .

$y = \frac{0}{0 - 6 \cdot 0 - 9}$

Step 2.2.4.2.3

Multiply $- 6$ by $0$ .

$y = \frac{0}{0 + 0 - 9}$

Step 2.2.4.2.4

Add $0$ and $0$ .

$y = \frac{0}{0 - 9}$

Step 2.2.4.2.5

Subtract $9$ from $0$ .

$y = \frac{0}{- 9}$

Step 2.2.4.3

Divide $0$ by $- 9$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 3, 0)$

y-intercept(s): $(0, 0)$

Step 4