Find the x and y Intercepts f(x)=-2(5x+2)(2x^2+6)(x-2)

$f (x) = - 2 (5 x + 2) (2 x^{2} + 6) (x - 2)$

Find the x-intercepts.

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

$0 = - 2 (5 x + 2) (2 x^{2} + 6) (x - 2)$

Solve the equation.

Rewrite the equation as $- 2 (5 x + 2) (2 x^{2} + 6) (x - 2) = 0$ .

$- 2 (5 x + 2) (2 x^{2} + 6) (x - 2) = 0$

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

$5 x + 2 = 0$

$2 x^{2} + 6 = 0$

$x - 2 = 0$

Set $5 x + 2$ equal to $0$ and solve for $x$ .

Set $5 x + 2$ equal to $0$ .

$5 x + 2 = 0$

Solve $5 x + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Divide each term in $5 x = - 2$ by $5$ and simplify.

Divide each term in $5 x = - 2$ by $5$ .

$\frac{5 x}{5} = \frac{- 2}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2}{5}$

Divide $x$ by $1$ .

$x = \frac{- 2}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Set $2 x^{2} + 6$ equal to $0$ and solve for $x$ .

Set $2 x^{2} + 6$ equal to $0$ .

$2 x^{2} + 6 = 0$

Solve $2 x^{2} + 6 = 0$ for $x$ .

Subtract $6$ from both sides of the equation.

$2 x^{2} = - 6$

Divide each term in $2 x^{2} = - 6$ by $2$ and simplify.

Divide each term in $2 x^{2} = - 6$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{- 6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{- 6}{2}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 6}{2}$

Simplify the right side.

Divide $- 6$ by $2$ .

$x^{2} = - 3$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 3}$

Simplify $\pm \sqrt{- 3}$ .

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 (3)}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{3}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{3}, - i \sqrt{3}$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $- 2 (5 x + 2) (2 x^{2} + 6) (x - 2) = 0$ true.

$x = - \frac{2}{5}, i \sqrt{3}, - i \sqrt{3}, 2$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{2}{5}, 0), (2, 0)$

Find the y-intercepts.

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

$y = - 2 (5 (0) + 2) (2 {(0)}^{2} + 6) ((0) - 2)$

Solve the equation.

Remove parentheses.

$y = - 2 (5 (0) + 2) (2 \cdot 0^{2} + 6) ((0) - 2)$

Remove parentheses.

$y = - 2 (5 (0) + 2) (2 \cdot 0^{2} + 6) (0 - 2)$

Remove parentheses.

$y = - 2 (5 (0) + 2) (2 {(0)}^{2} + 6) ((0) - 2)$

Simplify $- 2 (5 (0) + 2) (2 {(0)}^{2} + 6) ((0) - 2)$ .

Simplify the expression.

Multiply $5$ by $0$ .

$y = - 2 (0 + 2) (2 {(0)}^{2} + 6) ((0) - 2)$

Add $0$ and $2$ .

$y = - 2 \cdot 2 (2 {(0)}^{2} + 6) ((0) - 2)$

Multiply $- 2$ by $2$ .

$y = - 4 (2 {(0)}^{2} + 6) ((0) - 2)$

Simplify each term.

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

$y = - 4 (2 \cdot 0 + 6) ((0) - 2)$

Multiply $2$ by $0$ .

$y = - 4 (0 + 6) ((0) - 2)$

Simplify the expression.

Add $0$ and $6$ .

$y = - 4 \cdot 6 ((0) - 2)$

Multiply $- 4$ by $6$ .

$y = - 24 ((0) - 2)$

Subtract $2$ from $0$ .

$y = - 24 \cdot - 2$

Multiply $- 24$ by $- 2$ .

$y = 48$

y-intercept(s) in point form.

y-intercept(s): $(0, 48)$

List the intersections.

x-intercept(s): $(- \frac{2}{5}, 0), (2, 0)$

y-intercept(s): $(0, 48)$

Find the x and y Intercepts f(x)=-2(5x+2)(2x^2+6)(x-2)

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