Find the x and y Intercepts f(x)=-2x^3+10x^2+48x

$f (x) = - 2 x^{3} + 10 x^{2} + 48 x$

Step 1

Find the x-intercepts.

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

$0 = - 2 x^{3} + 10 x^{2} + 48 x$

Solve the equation.

Rewrite the equation as $- 2 x^{3} + 10 x^{2} + 48 x = 0$ .

$- 2 x^{3} + 10 x^{2} + 48 x = 0$

Factor the left side of the equation.

Factor $- 2 x$ out of $- 2 x^{3} + 10 x^{2} + 48 x$ .

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

$- 2 x \cdot x^{2} + 10 x^{2} + 48 x = 0$

Factor $- 2 x$ out of $10 x^{2}$ .

$- 2 x \cdot x^{2} - 2 x (- 5 x) + 48 x = 0$

Factor $- 2 x$ out of $48 x$ .

$- 2 x \cdot x^{2} - 2 x (- 5 x) - 2 x \cdot - 24 = 0$

Factor $- 2 x$ out of $- 2 x (x^{2}) - 2 x (- 5 x)$ .

$- 2 x (x^{2} - 5 x) - 2 x \cdot - 24 = 0$

Factor $- 2 x$ out of $- 2 x (x^{2} - 5 x) - 2 x (- 24)$ .

$- 2 x (x^{2} - 5 x - 24) = 0$

Factor.

Factor $x^{2} - 5 x - 24$ 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 $- 24$ and whose sum is $- 5$ .

$- 8, 3$

Write the factored form using these integers.

$- 2 x ((x - 8) (x + 3)) = 0$

Remove unnecessary parentheses.

$- 2 x (x - 8) (x + 3) = 0$

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

$x = 0$

$x - 8 = 0$

$x + 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $- 2 x (x - 8) (x + 3) = 0$ true.

$x = 0, 8, - 3$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (8, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

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

$y = - 2 {(0)}^{3} + 10 {(0)}^{2} + 48 (0)$

Solve the equation.

Remove parentheses.

$y = - 2 \cdot 0^{3} + 10 {(0)}^{2} + 48 (0)$

Remove parentheses.

$y = - 2 \cdot 0^{3} + 10 \cdot 0^{2} + 48 (0)$

Remove parentheses.

$y = - 2 {(0)}^{3} + 10 {(0)}^{2} + 48 (0)$

Simplify $- 2 {(0)}^{3} + 10 {(0)}^{2} + 48 (0)$ .

Simplify each term.

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

$y = - 2 \cdot 0 + 10 {(0)}^{2} + 48 (0)$

Multiply $- 2$ by $0$ .

$y = 0 + 10 {(0)}^{2} + 48 (0)$

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

$y = 0 + 10 \cdot 0 + 48 (0)$

Multiply $10$ by $0$ .

$y = 0 + 0 + 48 (0)$

Multiply $48$ 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): $(0, 0), (8, 0), (- 3, 0)$

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

Step 4

Find the x and y Intercepts f(x)=-2x^3+10x^2+48x

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