Find the x and y Intercepts (x^3-16x)/(-4x^2+4x+24)

$\frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$

Step 1

Write $\frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$ as an equation.

$y = \frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$

Step 2

Find the x-intercepts.

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

$0 = \frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$

Solve the equation.

Set the numerator equal to zero.

$x^{3} - 16 x = 0$

Solve the equation for $x$ .

Factor the left side of the equation.

Factor $x$ out of $x^{3} - 16 x$ .

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

$x \cdot x^{2} - 16 x = 0$

Factor $x$ out of $- 16 x$ .

$x \cdot x^{2} + x \cdot - 16 = 0$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 16$ .

$x (x^{2} - 16) = 0$

Rewrite $16$ as $4^{2}$ .

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

Factor.

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 $b = 4$ .

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

Remove unnecessary parentheses.

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

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

$x = 0$

$x + 4 = 0$

$x - 4 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

$x = 0, - 4, 4$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 4, 0), (4, 0)$

Step 3

Find the y-intercepts.

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

$y = \frac{{(0)}^{3} - 16 \cdot 0}{- 4 {(0)}^{2} + 4 (0) + 24}$

Solve the equation.

Remove parentheses.

$y = \frac{0^{3} - 16 \cdot 0}{- 4 {(0)}^{2} + 4 (0) + 24}$

Remove parentheses.

$y = \frac{0^{3} - 16 \cdot 0}{- 4 \cdot 0^{2} + 4 (0) + 24}$

Remove parentheses.

$y = \frac{{(0)}^{3} - 16 \cdot 0}{- 4 {(0)}^{2} + 4 (0) + 24}$

Simplify $\frac{{(0)}^{3} - 16 \cdot 0}{- 4 {(0)}^{2} + 4 (0) + 24}$ .

Simplify the numerator.

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

$y = \frac{0 - 16 \cdot 0}{- 4 \cdot 0^{2} + 4 (0) + 24}$

Multiply $- 16$ by $0$ .

$y = \frac{0 + 0}{- 4 \cdot 0^{2} + 4 (0) + 24}$

Add $0$ and $0$ .

$y = \frac{0}{- 4 \cdot 0^{2} + 4 (0) + 24}$

Simplify the denominator.

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

$y = \frac{0}{- 4 \cdot 0 + 4 (0) + 24}$

Multiply $- 4$ by $0$ .

$y = \frac{0}{0 + 4 (0) + 24}$

Multiply $4$ by $0$ .

$y = \frac{0}{0 + 0 + 24}$

Add $0$ and $0$ .

$y = \frac{0}{0 + 24}$

Add $0$ and $24$ .

$y = \frac{0}{24}$

Divide $0$ by $24$ .

$y = 0$

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $(0, 0), (- 4, 0), (4, 0)$

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

Step 5

Find the x and y Intercepts (x^3-16x)/(-4x^2+4x+24)

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