Find the x and y Intercepts f(x)=(x+6)(5x^2+2)(x-3)

$f (x) = (x + 6) (5 x^{2} + 2) (x - 3)$

Find the x-intercepts.

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

$0 = (x + 6) (5 x^{2} + 2) (x - 3)$

Solve the equation.

Rewrite the equation as $(x + 6) (5 x^{2} + 2) (x - 3) = 0$ .

$(x + 6) (5 x^{2} + 2) (x - 3) = 0$

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

$x + 6 = 0$

$5 x^{2} + 2 = 0$

$x - 3 = 0$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

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

Set $5 x^{2} + 2$ equal to $0$ .

$5 x^{2} + 2 = 0$

Solve $5 x^{2} + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$5 x^{2} = - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

Simplify $\pm \sqrt{- \frac{2}{5}}$ .

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

$x = \pm \sqrt{i^{2} \frac{2}{5}}$

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{2}{5}}$

Rewrite $\sqrt{\frac{2}{5}}$ as $\frac{\sqrt{2}}{\sqrt{5}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{5}}$

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm i (\frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{2 \cdot 5}}{5}$

Multiply $2$ by $5$ .

$x = \pm i \frac{\sqrt{10}}{5}$

Combine $i$ and $\frac{\sqrt{10}}{5}$ .

$x = \pm \frac{i \sqrt{10}}{5}$

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.

$x = \frac{i \sqrt{10}}{5}$

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

$x = - \frac{i \sqrt{10}}{5}$

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

$x = \frac{i \sqrt{10}}{5}, - \frac{i \sqrt{10}}{5}$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $(x + 6) (5 x^{2} + 2) (x - 3) = 0$ true.

$x = - 6, \frac{i \sqrt{10}}{5}, - \frac{i \sqrt{10}}{5}, 3$

x-intercept(s) in point form.

x-intercept(s): $(- 6, 0), (3, 0)$

Find the y-intercepts.

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

$y = ((0) + 6) (5 {(0)}^{2} + 2) ((0) - 3)$

Solve the equation.

Remove parentheses.

$y = (0 + 6) (5 {(0)}^{2} + 2) ((0) - 3)$

Remove parentheses.

$y = (0 + 6) (5 \cdot 0^{2} + 2) ((0) - 3)$

Remove parentheses.

$y = (0 + 6) (5 \cdot 0^{2} + 2) (0 - 3)$

Remove parentheses.

$y = ((0) + 6) (5 {(0)}^{2} + 2) ((0) - 3)$

Simplify $((0) + 6) (5 {(0)}^{2} + 2) ((0) - 3)$ .

Add $0$ and $6$ .

$y = 6 (5 {(0)}^{2} + 2) ((0) - 3)$

Simplify each term.

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

$y = 6 (5 \cdot 0 + 2) ((0) - 3)$

Multiply $5$ by $0$ .

$y = 6 (0 + 2) ((0) - 3)$

Simplify the expression.

Add $0$ and $2$ .

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

Multiply $6$ by $2$ .

$y = 12 ((0) - 3)$

Subtract $3$ from $0$ .

$y = 12 \cdot - 3$

Multiply $12$ by $- 3$ .

$y = - 36$

y-intercept(s) in point form.

y-intercept(s): $(0, - 36)$

List the intersections.

x-intercept(s): $(- 6, 0), (3, 0)$

y-intercept(s): $(0, - 36)$

Find the x and y Intercepts f(x)=(x+6)(5x^2+2)(x-3)

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