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

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

Find the x-intercepts.

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

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

Solve the equation.

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

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

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

$x + 3 = 0$

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

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.

$x = - 3$

Set $3 x^{2} + 2$ equal to $0$ and solve for $x$ .

Set $3 x^{2} + 2$ equal to $0$ .

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

Solve $3 x^{2} + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$3 x^{2} = - 2$

Divide each term in $3 x^{2} = - 2$ by $3$ and simplify.

Divide each term in $3 x^{2} = - 2$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

$x = \pm \sqrt{- \frac{2}{3}}$

Simplify $\pm \sqrt{- \frac{2}{3}}$ .

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{2}{3}}$

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{2}{3}}$

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{3}}$

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{2 \cdot 3}}{3}$

Multiply $2$ by $3$ .

$x = \pm i \frac{\sqrt{6}}{3}$

Combine $i$ and $\frac{\sqrt{6}}{3}$ .

$x = \pm \frac{i \sqrt{6}}{3}$

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.

$x = \frac{i \sqrt{6}}{3}$

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

$x = - \frac{i \sqrt{6}}{3}$

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

$x = \frac{i \sqrt{6}}{3}, - \frac{i \sqrt{6}}{3}$

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

$x = - 3, \frac{i \sqrt{6}}{3}, - \frac{i \sqrt{6}}{3}$

x-intercept(s) in point form.

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

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

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

Simplify the expression.

Add $0$ and $3$ .

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

Multiply $2$ by $3$ .

$y = 6 (3 {(0)}^{2} + 2)$

Simplify each term.

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

$y = 6 (3 \cdot 0 + 2)$

Multiply $3$ by $0$ .

$y = 6 (0 + 2)$

Simplify the expression.

Add $0$ and $2$ .

$y = 6 \cdot 2$

Multiply $6$ by $2$ .

$y = 12$

y-intercept(s) in point form.

y-intercept(s): $(0, 12)$

List the intersections.

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

y-intercept(s): $(0, 12)$

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

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