Find Where Undefined/Discontinuous f(x)=(x^3-5x^2-16x+80)/(x^2+x-30)

$f (x) = \frac{x^{3} - 5 x^{2} - 16 x + 80}{x^{2} + x - 30}$

Step 1

Set the denominator in $\frac{x^{3} - 5 x^{2} - 16 x + 80}{x^{2} + x - 30}$ equal to $0$ to find where the expression is undefined.

$x^{2} + x - 30 = 0$

Step 2

Solve for $x$ .

Factor $x^{2} + x - 30$ 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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Write the factored form using these integers.

$(x - 5) (x + 6) = 0$

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

$x - 5 = 0$

$x + 6 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x - 5) (x + 6) = 0$ true.

$x = 5, - 6$

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 $0$ .

$x = - 6, x = 5$

Step 4

Find Where Undefined/Discontinuous f(x)=(x^3-5x^2-16x+80)/(x^2+x-30)

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