Precalculus Examples

$\frac{x^{3} - 3 x^{2} - 4 x}{x - 4}$

Step 1

Write $\frac{x^{3} - 3 x^{2} - 4 x}{x - 4}$ as an equation.

$y = \frac{x^{3} - 3 x^{2} - 4 x}{x - 4}$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = \frac{x^{3} - 3 x^{2} - 4 x}{x - 4}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$x^{3} - 3 x^{2} - 4 x = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

Factor the left side of the equation.

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Step 2.2.2.1.1

Factor $x$ out of $x^{3} - 3 x^{2} - 4 x$ .

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Step 2.2.2.1.1.1

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

$x \cdot x^{2} - 3 x^{2} - 4 x = 0$

Step 2.2.2.1.1.2

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

$x \cdot x^{2} + x (- 3 x) - 4 x = 0$

Step 2.2.2.1.1.3

Factor $x$ out of $- 4 x$ .

$x \cdot x^{2} + x (- 3 x) + x \cdot - 4 = 0$

Step 2.2.2.1.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 3 x)$ .

$x (x^{2} - 3 x) + x \cdot - 4 = 0$

Step 2.2.2.1.1.5

Factor $x$ out of $x (x^{2} - 3 x) + x \cdot - 4$ .

$x (x^{2} - 3 x - 4) = 0$

Step 2.2.2.1.2

Factor.

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Step 2.2.2.1.2.1

Factor $x^{2} - 3 x - 4$ using the AC method.

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Step 2.2.2.1.2.1.1

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 $- 3$ .

$- 4, 1$

Step 2.2.2.1.2.1.2

Write the factored form using these integers.

$x ((x - 4) (x + 1)) = 0$

Step 2.2.2.1.2.2

Remove unnecessary parentheses.

$x (x - 4) (x + 1) = 0$

Step 2.2.2.2

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

$x = 0$

$x - 4 = 0$

$x + 1 = 0$

Step 2.2.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.2.2.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.2.2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.2.2.5

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.2.2.5.1

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

$x + 1 = 0$

Step 2.2.2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.2.2.6

The final solution is all the values that make $x (x - 4) (x + 1) = 0$ true.

$x = 0, 4, - 1$

Step 2.2.3

Exclude the solutions that do not make $0 = \frac{x^{3} - 3 x^{2} - 4 x}{x - 4}$ true.

$x = 0, - 1$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 1, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = \frac{{(0)}^{3} - 3 {(0)}^{2} - 4 \cdot 0}{(0) - 4}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \frac{0^{3} - 3 {(0)}^{2} - 4 \cdot 0}{(0) - 4}$

Step 3.2.2

Remove parentheses.

$y = \frac{0^{3} - 3 \cdot 0^{2} - 4 \cdot 0}{(0) - 4}$

Step 3.2.3

Remove parentheses.

$y = \frac{0^{3} - 3 \cdot 0^{2} - 4 \cdot 0}{0 - 4}$

Step 3.2.4

Remove parentheses.

$y = \frac{{(0)}^{3} - 3 {(0)}^{2} - 4 \cdot 0}{(0) - 4}$

Step 3.2.5

Simplify $\frac{{(0)}^{3} - 3 {(0)}^{2} - 4 \cdot 0}{(0) - 4}$ .

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Step 3.2.5.1

Simplify the numerator.

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Step 3.2.5.1.1

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

$y = \frac{0 - 3 \cdot 0^{2} - 4 \cdot 0}{0 - 4}$

Step 3.2.5.1.2

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

$y = \frac{0 - 3 \cdot 0 - 4 \cdot 0}{0 - 4}$

Step 3.2.5.1.3

Multiply $- 3$ by $0$ .

$y = \frac{0 + 0 - 4 \cdot 0}{0 - 4}$

Step 3.2.5.1.4

Multiply $- 4$ by $0$ .

$y = \frac{0 + 0 + 0}{0 - 4}$

Step 3.2.5.1.5

Add $0 + 0$ and $0$ .

$y = \frac{0 + 0}{0 - 4}$

Step 3.2.5.1.6

Add $0$ and $0$ .

$y = \frac{0}{0 - 4}$

Step 3.2.5.2

Simplify the expression.

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Step 3.2.5.2.1

Subtract $4$ from $0$ .

$y = \frac{0}{- 4}$

Step 3.2.5.2.2

Divide $0$ by $- 4$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $(0, 0), (- 1, 0)$

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

Step 5