Find the x and y Intercepts f(x)=3x(x+4)^2(x^2-7)(6x+1)(x^2+4)

$f (x) = 3 x {(x + 4)}^{2} (x^{2} - 7) (6 x + 1) (x^{2} + 4)$

Step 1

Find the x-intercepts.

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

$0 = 3 x {(x + 4)}^{2} (x^{2} - 7) (6 x + 1) (x^{2} + 4)$

Solve the equation.

Rewrite the equation as $3 x {(x + 4)}^{2} (x^{2} - 7) (6 x + 1) (x^{2} + 4) = 0$ .

$3 x {(x + 4)}^{2} (x^{2} - 7) (6 x + 1) (x^{2} + 4) = 0$

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

$x = 0$

${(x + 4)}^{2} = 0$

$x^{2} - 7 = 0$

$6 x + 1 = 0$

$x^{2} + 4 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set ${(x + 4)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x + 4)}^{2}$ equal to $0$ .

${(x + 4)}^{2} = 0$

Solve ${(x + 4)}^{2} = 0$ for $x$ .

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Set $x^{2} - 7$ equal to $0$ and solve for $x$ .

Set $x^{2} - 7$ equal to $0$ .

$x^{2} - 7 = 0$

Solve $x^{2} - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$x^{2} = 7$

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

$x = \pm \sqrt{7}$

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.

$x = \sqrt{7}$

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

$x = - \sqrt{7}$

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

$x = \sqrt{7}, - \sqrt{7}$

Set $6 x + 1$ equal to $0$ and solve for $x$ .

Set $6 x + 1$ equal to $0$ .

$6 x + 1 = 0$

Solve $6 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$6 x = - 1$

Divide each term in $6 x = - 1$ by $6$ and simplify.

Divide each term in $6 x = - 1$ by $6$ .

$\frac{6 x}{6} = \frac{- 1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 1}{6}$

Divide $x$ by $1$ .

$x = \frac{- 1}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{6}$

Set $x^{2} + 4$ equal to $0$ and solve for $x$ .

Set $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Solve $x^{2} + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

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

$x = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

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

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

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

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

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

$x = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$x = \pm 2 i$

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.

$x = 2 i$

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

$x = - 2 i$

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

$x = 2 i, - 2 i$

The final solution is all the values that make $3 x {(x + 4)}^{2} (x^{2} - 7) (6 x + 1) (x^{2} + 4) = 0$ true.

$x = 0, - 4, \sqrt{7}, - \sqrt{7}, - \frac{1}{6}, 2 i, - 2 i$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 4, 0), (\sqrt{7}, 0), (- \sqrt{7}, 0), (- \frac{1}{6}, 0)$

Step 2

Find the y-intercepts.

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

$y = 3 (0) {((0) + 4)}^{2} ({(0)}^{2} - 7) (6 (0) + 1) ({(0)}^{2} + 4)$

Solve the equation.

Remove parentheses.

$y = 3 (0) {(0 + 4)}^{2} ({(0)}^{2} - 7) (6 (0) + 1) ({(0)}^{2} + 4)$

Remove parentheses.

$y = 3 (0) {(0 + 4)}^{2} (0^{2} - 7) (6 (0) + 1) ({(0)}^{2} + 4)$

Remove parentheses.

$y = 3 (0) {(0 + 4)}^{2} (0^{2} - 7) (6 (0) + 1) (0^{2} + 4)$

Remove parentheses.

$y = 3 (0) {((0) + 4)}^{2} ({(0)}^{2} - 7) (6 (0) + 1) ({(0)}^{2} + 4)$

Simplify $3 (0) {((0) + 4)}^{2} ({(0)}^{2} - 7) (6 (0) + 1) ({(0)}^{2} + 4)$ .

Multiply $3$ by $0$ .

$y = 0 {((0) + 4)}^{2} ({(0)}^{2} - 7) (6 (0) + 1) ({(0)}^{2} + 4)$

Add $0$ and $4$ .

$y = 0 \cdot 4^{2} ({(0)}^{2} - 7) (6 (0) + 1) ({(0)}^{2} + 4)$

Raise $4$ to the power of $2$ .

$y = 0 \cdot 16 ({(0)}^{2} - 7) (6 (0) + 1) ({(0)}^{2} + 4)$

Multiply $0$ by $16$ .

$y = 0 ({(0)}^{2} - 7) (6 (0) + 1) ({(0)}^{2} + 4)$

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

$y = 0 (0 - 7) (6 (0) + 1) ({(0)}^{2} + 4)$

Subtract $7$ from $0$ .

$y = 0 \cdot - 7 (6 (0) + 1) ({(0)}^{2} + 4)$

Multiply $0$ by $- 7$ .

$y = 0 (6 (0) + 1) ({(0)}^{2} + 4)$

Multiply $6$ by $0$ .

$y = 0 (0 + 1) ({(0)}^{2} + 4)$

Add $0$ and $1$ .

$y = 0 \cdot 1 ({(0)}^{2} + 4)$

Multiply $0$ by $1$ .

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

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

$y = 0 (0 + 4)$

Add $0$ and $4$ .

$y = 0 \cdot 4$

Multiply $0$ by $4$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 4, 0), (\sqrt{7}, 0), (- \sqrt{7}, 0), (- \frac{1}{6}, 0)$

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

Step 4

Find the x and y Intercepts f(x)=3x(x+4)^2(x^2-7)(6x+1)(x^2+4)

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