To determine for which values [tex]\(x\)[/tex] makes the given rational expression [tex]\(\frac{x + 5}{3x^2 - 3}\)[/tex] undefined, we need to identify when the denominator equals zero. The rational expression is undefined wherever the denominator is zero because division by zero is undefined.
Let's set the denominator equal to zero and solve for [tex]\(x\)[/tex]:
1. Start with the denominator:
[tex]\[
3x^2 - 3 = 0
\][/tex]
2. Factor out the common factor:
[tex]\[
3(x^2 - 1) = 0
\][/tex]
3. Now, solve the equation inside the parentheses:
[tex]\[
x^2 - 1 = 0
\][/tex]
4. This is a difference of squares, which factors into:
[tex]\[
(x - 1)(x + 1) = 0
\][/tex]
5. Setting each factor equal to zero gives us:
[tex]\[
x - 1 = 0 \quad \text{or} \quad x + 1 = 0
\][/tex]
6. Solving these equations for [tex]\(x\)[/tex]:
[tex]\[
x = 1 \quad \text{or} \quad x = -1
\][/tex]
Therefore, the rational expression [tex]\(\frac{x + 5}{3x^2 - 3}\)[/tex] is undefined for [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex].
Now, let's match these results with the given options:
A. 3
B. -1
C. 5
D. -5
E. 1
F. -3
The values of [tex]\(x\)[/tex] that make the rational expression undefined are [tex]\(x = 1\)[/tex] (Option E) and [tex]\(x = -1\)[/tex] (Option B). Hence, the correct options are B and E.
