Find the x and y Intercepts f(x)=-3x^3-21x^2-30x

$f (x) = - 3 x^{3} - 21 x^{2} - 30 x$

Step 1

Find the x-intercepts.

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

$0 = - 3 x^{3} - 21 x^{2} - 30 x$

Solve the equation.

Rewrite the equation as $- 3 x^{3} - 21 x^{2} - 30 x = 0$ .

$- 3 x^{3} - 21 x^{2} - 30 x = 0$

Factor the left side of the equation.

Factor $- 3 x$ out of $- 3 x^{3} - 21 x^{2} - 30 x$ .

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

$- 3 x \cdot x^{2} - 21 x^{2} - 30 x = 0$

Factor $- 3 x$ out of $- 21 x^{2}$ .

$- 3 x \cdot x^{2} - 3 x (7 x) - 30 x = 0$

Factor $- 3 x$ out of $- 30 x$ .

$- 3 x \cdot x^{2} - 3 x (7 x) - 3 x \cdot 10 = 0$

Factor $- 3 x$ out of $- 3 x (x^{2}) - 3 x (7 x)$ .

$- 3 x (x^{2} + 7 x) - 3 x \cdot 10 = 0$

Factor $- 3 x$ out of $- 3 x (x^{2} + 7 x) - 3 x (10)$ .

$- 3 x (x^{2} + 7 x + 10) = 0$

Factor.

Factor $x^{2} + 7 x + 10$ 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 $10$ and whose sum is $7$ .

$2, 5$

Write the factored form using these integers.

$- 3 x ((x + 2) (x + 5)) = 0$

Remove unnecessary parentheses.

$- 3 x (x + 2) (x + 5) = 0$

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

$x = 0$

$x + 2 = 0$

$x + 5 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $- 3 x (x + 2) (x + 5) = 0$ true.

$x = 0, - 2, - 5$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 2, 0), (- 5, 0)$

Step 2

Find the y-intercepts.

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

$y = - 3 {(0)}^{3} - 21 {(0)}^{2} - 30 \cdot 0$

Solve the equation.

Remove parentheses.

$y = - 3 \cdot 0^{3} - 21 {(0)}^{2} - 30 \cdot 0$

Remove parentheses.

$y = - 3 \cdot 0^{3} - 21 \cdot 0^{2} - 30 \cdot 0$

Remove parentheses.

$y = - 3 {(0)}^{3} - 21 {(0)}^{2} - 30 \cdot 0$

Simplify $- 3 {(0)}^{3} - 21 {(0)}^{2} - 30 \cdot 0$ .

Simplify each term.

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

$y = - 3 \cdot 0 - 21 {(0)}^{2} - 30 \cdot 0$

Multiply $- 3$ by $0$ .

$y = 0 - 21 {(0)}^{2} - 30 \cdot 0$

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

$y = 0 - 21 \cdot 0 - 30 \cdot 0$

Multiply $- 21$ by $0$ .

$y = 0 + 0 - 30 \cdot 0$

Multiply $- 30$ 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): $(0, 0), (- 2, 0), (- 5, 0)$

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

Step 4

Find the x and y Intercepts f(x)=-3x^3-21x^2-30x

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