Find the x and y Intercepts f(x)=-x^3-3x^2+54x

$f (x) = - x^{3} - 3 x^{2} + 54 x$

Step 1

Find the x-intercepts.

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

$0 = - x^{3} - 3 x^{2} + 54 x$

Solve the equation.

Rewrite the equation as $- x^{3} - 3 x^{2} + 54 x = 0$ .

$- x^{3} - 3 x^{2} + 54 x = 0$

Factor the left side of the equation.

Factor $- x$ out of $- x^{3} - 3 x^{2} + 54 x$ .

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

$- x \cdot x^{2} - 3 x^{2} + 54 x = 0$

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

$- x \cdot x^{2} - x (3 x) + 54 x = 0$

Factor $- x$ out of $54 x$ .

$- x \cdot x^{2} - x (3 x) - x \cdot - 54 = 0$

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

$- x (x^{2} + 3 x) - x \cdot - 54 = 0$

Factor $- x$ out of $- x (x^{2} + 3 x) - x (- 54)$ .

$- x (x^{2} + 3 x - 54) = 0$

Factor.

Factor $x^{2} + 3 x - 54$ using the AC method.

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 $- 54$ and whose sum is $3$ .

$- 6, 9$

Write the factored form using these integers.

$- x ((x - 6) (x + 9)) = 0$

Remove unnecessary parentheses.

$- x (x - 6) (x + 9) = 0$

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

$x = 0$

$x - 6 = 0$

$x + 9 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

The final solution is all the values that make $- x (x - 6) (x + 9) = 0$ true.

$x = 0, 6, - 9$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (6, 0), (- 9, 0)$

Step 2

Find the y-intercepts.

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

$y = - {(0)}^{3} - 3 {(0)}^{2} + 54 (0)$

Solve the equation.

Remove parentheses.

$y = - 0^{3} - 3 {(0)}^{2} + 54 (0)$

Remove parentheses.

$y = - 0^{3} - 3 \cdot 0^{2} + 54 (0)$

Remove parentheses.

$y = - {(0)}^{3} - 3 {(0)}^{2} + 54 (0)$

Simplify $- {(0)}^{3} - 3 {(0)}^{2} + 54 (0)$ .

Simplify each term.

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

$y = - 0 - 3 {(0)}^{2} + 54 (0)$

Multiply $- 1$ by $0$ .

$y = 0 - 3 {(0)}^{2} + 54 (0)$

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

$y = 0 - 3 \cdot 0 + 54 (0)$

Multiply $- 3$ by $0$ .

$y = 0 + 0 + 54 (0)$

Multiply $54$ by $0$ .

$y = 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (6, 0), (- 9, 0)$

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

Step 4

Find the x and y Intercepts f(x)=-x^3-3x^2+54x

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