To determine for which values of [tex]\( x \)[/tex] the rational expression [tex]\(\frac{x+8}{x^2-2x-24}\)[/tex] is undefined, we need to identify the values of [tex]\( x \)[/tex] that make the denominator zero.
Let's start by examining the denominator:
[tex]\[ x^2 - 2x - 24 \][/tex]
To find the values of [tex]\( x \)[/tex] that make this expression equal to zero, we start by factoring it. We look for two numbers that multiply to [tex]\(-24\)[/tex] (the constant term) and add up to [tex]\(-2\)[/tex] (the coefficient of the [tex]\( x \)[/tex] term).
Those two numbers are [tex]\( 6 \)[/tex] and [tex]\(-4\)[/tex], because:
[tex]\[ 6 \cdot (-4) = -24 \][/tex]
[tex]\[ 6 + (-4) = 2 \][/tex]
Thus, we can factor the quadratic expression as:
[tex]\[ x^2 - 2x - 24 = (x - 6)(x + 4) \][/tex]
Next, we find the values of [tex]\( x \)[/tex] that make each factor equal to zero:
1. [tex]\( x - 6 = 0 \)[/tex]
[tex]\[ x = 6 \][/tex]
2. [tex]\( x + 4 = 0 \)[/tex]
[tex]\[ x = -4 \][/tex]
Therefore, the rational expression [tex]\(\frac{x+8}{x^2-2x-24}\)[/tex] is undefined for [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].
So, the correct answers are:
B. -4
C. 6
