Find Where Undefined/Discontinuous (z^2+2z+1)/(z^2-6z-7)

$\frac{z^{2} + 2 z + 1}{z^{2} - 6 z - 7}$

Step 1

Set the denominator in $\frac{z^{2} + 2 z + 1}{z^{2} - 6 z - 7}$ equal to $0$ to find where the expression is undefined.

$z^{2} - 6 z - 7 = 0$

Step 2

Solve for $z$ .

Factor $z^{2} - 6 z - 7$ 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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Write the factored form using these integers.

$(z - 7) (z + 1) = 0$

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

$z - 7 = 0$

$z + 1 = 0$

Set $z - 7$ equal to $0$ and solve for $z$ .

Set $z - 7$ equal to $0$ .

$z - 7 = 0$

Add $7$ to both sides of the equation.

$z = 7$

Set $z + 1$ equal to $0$ and solve for $z$ .

Set $z + 1$ equal to $0$ .

$z + 1 = 0$

Subtract $1$ from both sides of the equation.

$z = - 1$

The final solution is all the values that make $(z - 7) (z + 1) = 0$ true.

$z = 7, - 1$

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 $0$ .

$z = - 1, z = 7$

Step 4

Find Where Undefined/Discontinuous (z^2+2z+1)/(z^2-6z-7)

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