Find Where Undefined/Discontinuous (z^2+12z+27)/(z^3+3z^2-54z)

$\frac{z^{2} + 12 z + 27}{z^{3} + 3 z^{2} - 54 z}$

Step 1

Set the denominator in $\frac{z^{2} + 12 z + 27}{z^{3} + 3 z^{2} - 54 z}$ equal to $0$ to find where the expression is undefined.

$z^{3} + 3 z^{2} - 54 z = 0$

Step 2

Solve for $z$ .

Factor the left side of the equation.

Factor $z$ out of $z^{3} + 3 z^{2} - 54 z$ .

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

$z \cdot z^{2} + 3 z^{2} - 54 z = 0$

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

$z \cdot z^{2} + z (3 z) - 54 z = 0$

Factor $z$ out of $- 54 z$ .

$z \cdot z^{2} + z (3 z) + z \cdot - 54 = 0$

Factor $z$ out of $z \cdot z^{2} + z (3 z)$ .

$z (z^{2} + 3 z) + z \cdot - 54 = 0$

Factor $z$ out of $z (z^{2} + 3 z) + z \cdot - 54$ .

$z (z^{2} + 3 z - 54) = 0$

Factor.

Factor $z^{2} + 3 z - 54$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 54$ and whose sum is $3$ .

$- 6, 9$

Write the factored form using these integers.

$z ((z - 6) (z + 9)) = 0$

Remove unnecessary parentheses.

$z (z - 6) (z + 9) = 0$

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

$z = 0$

$z - 6 = 0$

$z + 9 = 0$

Set $z$ equal to $0$ .

$z = 0$

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

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

$z - 6 = 0$

Add $6$ to both sides of the equation.

$z = 6$

Set $z + 9$ equal to $0$ and solve for $z$ .

Set $z + 9$ equal to $0$ .

$z + 9 = 0$

Subtract $9$ from both sides of the equation.

$z = - 9$

The final solution is all the values that make $z (z - 6) (z + 9) = 0$ true.

$z = 0, 6, - 9$

Step 3

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 $0$ .

$z = - 9, z = 0, z = 6$

Step 4

Find Where Undefined/Discontinuous (z^2+12z+27)/(z^3+3z^2-54z)

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