Find the x and y Intercepts f(x)=-2x(2x^2+2)(2x-3)(5x+2)

$f (x) = - 2 x (2 x^{2} + 2) (2 x - 3) (5 x + 2)$

Step 1

Find the x-intercepts.

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

$0 = - 2 x (2 x^{2} + 2) (2 x - 3) (5 x + 2)$

Solve the equation.

Rewrite the equation as $- 2 x (2 x^{2} + 2) (2 x - 3) (5 x + 2) = 0$ .

$- 2 x (2 x^{2} + 2) (2 x - 3) (5 x + 2) = 0$

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

$x = 0$

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

$2 x - 3 = 0$

$5 x + 2 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

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

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

Subtract $2$ from both sides of the equation.

$2 x^{2} = - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Divide $- 2$ by $2$ .

$x^{2} = - 1$

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

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

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

$x = \pm i$

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.

$x = i$

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

$x = - i$

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

$x = i, - i$

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Divide $x$ by $1$ .

$x = \frac{3}{2}$

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.

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

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

$x = 0, i, - i, \frac{3}{2}, - \frac{2}{5}$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{3}{2}, 0), (- \frac{2}{5}, 0)$

Step 2

Find the y-intercepts.

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

$y = - 2 \cdot 0 (2 {(0)}^{2} + 2) (2 (0) - 3) (5 (0) + 2)$

Solve the equation.

Remove parentheses.

$y = - 2 \cdot (0 (2 \cdot 0^{2} + 2) (2 (0) - 3) (5 (0) + 2))$

Remove parentheses.

$y = - 2 \cdot (0 (2 {(0)}^{2} + 2) (2 (0) - 3) (5 (0) + 2))$

Simplify $- 2 \cdot (0 (2 {(0)}^{2} + 2) (2 (0) - 3) (5 (0) + 2))$ .

Simplify each term.

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

$y = - 2 \cdot (0 (2 \cdot 0 + 2) (2 (0) - 3) (5 (0) + 2))$

Multiply $2$ by $0$ .

$y = - 2 \cdot (0 (0 + 2) (2 (0) - 3) (5 (0) + 2))$

Simplify the expression.

Add $0$ and $2$ .

$y = - 2 \cdot (0 \cdot 2 (2 (0) - 3) (5 (0) + 2))$

Multiply $0$ by $2$ .

$y = - 2 \cdot (0 (2 (0) - 3) (5 (0) + 2))$

Multiply $2$ by $0$ .

$y = - 2 \cdot (0 (0 - 3) (5 (0) + 2))$

Subtract $3$ from $0$ .

$y = - 2 \cdot (0 \cdot - 3 (5 (0) + 2))$

Multiply $0$ by $- 3$ .

$y = - 2 \cdot (0 (5 (0) + 2))$

Multiply $5$ by $0$ .

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

Add $0$ and $2$ .

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

Multiply $- 2 (0 \cdot 2)$ .

Multiply $0$ by $2$ .

$y = - 2 \cdot 0$

Multiply $- 2$ by $0$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\frac{3}{2}, 0), (- \frac{2}{5}, 0)$

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

Step 4

Find the x and y Intercepts f(x)=-2x(2x^2+2)(2x-3)(5x+2)

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