Find the x and y Intercepts f(x)=3(x^2+16)(x-6)(x-6)(3x+2)

$f (x) = 3 (x^{2} + 16) (x - 6) (x - 6) (3 x + 2)$

Step 1

Find the x-intercepts.

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

$0 = 3 (x^{2} + 16) (x - 6) (x - 6) (3 x + 2)$

Solve the equation.

Rewrite the equation as $3 (x^{2} + 16) (x - 6) (x - 6) (3 x + 2) = 0$ .

$3 (x^{2} + 16) (x - 6) (x - 6) (3 x + 2) = 0$

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

$x^{2} + 16 = 0$

$x - 6 = 0$

$3 x + 2 = 0$

Set $x^{2} + 16$ equal to $0$ and solve for $x$ .

Set $x^{2} + 16$ equal to $0$ .

$x^{2} + 16 = 0$

Solve $x^{2} + 16 = 0$ for $x$ .

Subtract $16$ from both sides of the equation.

$x^{2} = - 16$

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

$x = \pm \sqrt{- 16}$

Simplify $\pm \sqrt{- 16}$ .

Rewrite $- 16$ as $- 1 (16)$ .

$x = \pm \sqrt{- 1 (16)}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{16}$

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

$x = \pm i \cdot \sqrt{16}$

Rewrite $16$ as $4^{2}$ .

$x = \pm i \cdot \sqrt{4^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 4$

Move $4$ to the left of $i$ .

$x = \pm 4 i$

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.

$x = 4 i$

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

$x = - 4 i$

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

$x = 4 i, - 4 i$

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

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

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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

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

$3 x + 2 = 0$

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

Subtract $2$ from both sides of the equation.

$3 x = - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

The final solution is all the values that make $3 (x^{2} + 16) (x - 6) (x - 6) (3 x + 2) = 0$ true.

$x = 4 i, - 4 i, 6, - \frac{2}{3}$

x-intercept(s) in point form.

x-intercept(s): $(6, 0), (- \frac{2}{3}, 0)$

Step 2

Find the y-intercepts.

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

$y = 3 ({(0)}^{2} + 16) ((0) - 6) ((0) - 6) (3 (0) + 2)$

Solve the equation.

Remove parentheses.

$y = 3 (0^{2} + 16) ((0) - 6) ((0) - 6) (3 (0) + 2)$

Remove parentheses.

$y = 3 (0^{2} + 16) (0 - 6) ((0) - 6) (3 (0) + 2)$

Remove parentheses.

$y = 3 (0^{2} + 16) (0 - 6) (0 - 6) (3 (0) + 2)$

Remove parentheses.

$y = 3 ({(0)}^{2} + 16) ((0) - 6) ((0) - 6) (3 (0) + 2)$

Simplify $3 ({(0)}^{2} + 16) ((0) - 6) ((0) - 6) (3 (0) + 2)$ .

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

$y = 3 (0 + 16) ((0) - 6) ((0) - 6) (3 (0) + 2)$

Add $0$ and $16$ .

$y = 3 \cdot 16 ((0) - 6) ((0) - 6) (3 (0) + 2)$

Multiply $3$ by $16$ .

$y = 48 ((0) - 6) ((0) - 6) (3 (0) + 2)$

Subtract $6$ from $0$ .

$y = 48 \cdot - 6 ((0) - 6) (3 (0) + 2)$

Multiply $48$ by $- 6$ .

$y = - 288 ((0) - 6) (3 (0) + 2)$

Subtract $6$ from $0$ .

$y = - 288 \cdot - 6 (3 (0) + 2)$

Multiply $- 288$ by $- 6$ .

$y = 1728 (3 (0) + 2)$

Multiply $3$ by $0$ .

$y = 1728 (0 + 2)$

Add $0$ and $2$ .

$y = 1728 \cdot 2$

Multiply $1728$ by $2$ .

$y = 3456$

y-intercept(s) in point form.

y-intercept(s): $(0, 3456)$

Step 3

List the intersections.

x-intercept(s): $(6, 0), (- \frac{2}{3}, 0)$

y-intercept(s): $(0, 3456)$

Step 4

Find the x and y Intercepts f(x)=3(x^2+16)(x-6)(x-6)(3x+2)

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