To add the rational expressions \( \frac{x^2 - 23x - 2}{3x - 2} + \frac{x + 5}{3x - 2} \), we first note that they have a common denominator of \( 3x - 2 \).
We can combine the numerators:
\[ \frac{(x^2 - 23x - 2) + (x + 5)}{3x - 2} \]
Now, simplify the numerator:
\[ x^2 - 23x - 2 + x + 5 = x^2 - 22x + 3 \]
So, we have:
\[ \frac{x^2 - 22x + 3}{3x - 2} \]
Now we can compare this expression with the given options.
It looks like none of the provided answers exactly match this result. However, if we further simplify \(x^2 - 22x + 3\), we can look for a suitable representation.
Now let's analyze the given options:
- \( \frac{x^2 + x + 7}{3x - 2} \)
- \( \frac{x^2 + x + 3}{3x - 2} \)
- \( \frac{x^3 + 3}{3x - 2} \)
- \( \frac{x^2 + x + 36}{6x - 4} \)
None of these options match \( \frac{x^2 - 22x + 3}{3x - 2} \) directly.
However, upon closer inspection, the presence of \(x\) in options (1) and (2) may lead to an error in calculations or assumptions.
Thus, based on the closest rational expression indicating a sum that might need to align with earlier context you meditated, the selection should ideally be derived from where the coefficients could align in some fashion with expectations halfway through rational addition.
Since there isn't an exact suitable expression, I'll conclude upon rechecking and conclude there could be a problem with any disjoint mismatches between \(x^2 - 22x + 3\) and all options.
From your provided options, none truly fulfills the correctly derived expression from these specific rational sums precisely unless further transformations of any specific numerical constants are aligned more finely.
Upon review from deduction traces, the right direct correspondence pattern brings forth to potentially align:
- Option 2 might be visually nearest, but logically direct checks downvalues versus the intended rationalization.
I would advise re-checking initial interpretable terms given to finalize an assertive spotting more adeptly to rational confines mapped accurately.
Nonetheless, I cannot assert definitively from provided selections without exacting clarity across each specified variant.
1 answer