Find Where Undefined/Discontinuous (6z^2-48z-120)/(3z^2+6z)*(14z^5+140z^4)/(4z^2-400)

$\frac{6 z^{2} - 48 z - 120}{3 z^{2} + 6 z} \cdot \frac{14 z^{5} + 140 z^{4}}{4 z^{2} - 400}$

Step 1

Set the denominator in $\frac{6 z^{2} - 48 z - 120}{3 z^{2} + 6 z}$ equal to $0$ to find where the expression is undefined.

$3 z^{2} + 6 z = 0$

Step 2

Solve for $z$ .

Factor $3 z$ out of $3 z^{2} + 6 z$ .

Factor $3 z$ out of $3 z^{2}$ .

$3 z (z) + 6 z = 0$

Factor $3 z$ out of $6 z$ .

$3 z (z) + 3 z (2) = 0$

Factor $3 z$ out of $3 z (z) + 3 z (2)$ .

$3 z (z + 2) = 0$

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

$z = 0$

$z + 2 = 0$

Set $z$ equal to $0$ .

$z = 0$

Set $z + 2$ equal to $0$ and solve for $z$ .

Set $z + 2$ equal to $0$ .

$z + 2 = 0$

Subtract $2$ from both sides of the equation.

$z = - 2$

The final solution is all the values that make $3 z (z + 2) = 0$ true.

$z = 0, - 2$

Step 3

Set the denominator in $\frac{14 z^{5} + 140 z^{4}}{4 z^{2} - 400}$ equal to $0$ to find where the expression is undefined.

$4 z^{2} - 400 = 0$

Step 4

Solve for $z$ .

Add $400$ to both sides of the equation.

$4 z^{2} = 400$

Divide each term in $4 z^{2} = 400$ by $4$ and simplify.

Divide each term in $4 z^{2} = 400$ by $4$ .

$\frac{4 z^{2}}{4} = \frac{400}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 z^{2}}{4} = \frac{400}{4}$

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

$z^{2} = \frac{400}{4}$

Simplify the right side.

Divide $400$ by $4$ .

$z^{2} = 100$

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

$z = \pm \sqrt{100}$

Simplify $\pm \sqrt{100}$ .

Rewrite $100$ as $10^{2}$ .

$z = \pm \sqrt{10^{2}}$

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

$z = \pm 10$

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.

$z = 10$

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

$z = - 10$

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

$z = 10, - 10$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$z = - 10, z = - 2, z = 0, z = 10$

Step 6

Find Where Undefined/Discontinuous (6z^2-48z-120)/(3z^2+6z)*(14z^5+140z^4)/(4z^2-400)

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