Question
Add the rational expressions to find the sum: x2−23x−2+x+53x−2
.(1 point)
Responses
x2+x+33x−2
Start Fraction x squared plus x plus 3 over 3 x minus 2 End Fraction
x2+x+36x−4
Start Fraction x squared plus x plus 3 over 6 x minus 4 End Fraction
x3+33x−2
Start Fraction x cubed plus 3 over 3 x minus 2 End Fraction
x2+x+73x−2
.(1 point)
Responses
x2+x+33x−2
Start Fraction x squared plus x plus 3 over 3 x minus 2 End Fraction
x2+x+36x−4
Start Fraction x squared plus x plus 3 over 6 x minus 4 End Fraction
x3+33x−2
Start Fraction x cubed plus 3 over 3 x minus 2 End Fraction
x2+x+73x−2
Answers
GPT-4o mini
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, \( 3x - 2 \).
Now we can combine the numerators:
\[
\frac{x^2 - 23x - 2 + x + 5}{3x - 2}
\]
Next, we simplify the numerator:
1. Combine like terms in the numerator:
\[
x^2 - 23x + x - 2 + 5 = x^2 - 22x + 3
\]
Now we have:
\[
\frac{x^2 - 22x + 3}{3x - 2}
\]
None of the provided answer options match our result exactly, but let's look at the closest format and determine which could potentially be correct:
1. **Option 1**: \( \frac{x^2 + x + 3}{3x - 2} \)
2. **Option 2**: \( \frac{x^2 + x + 3}{6x - 4} \)
3. **Option 3**: \( \frac{x^3 + 3}{3x - 2} \)
4. **Option 4**: \( \frac{x^2 + x + 7}{3x - 2} \)
Given our numerator is \(x^2 - 22x + 3\), the first and the fourth options do not fit, and the third option presents a cubic expression, which is not our case.
The second option simplifies with a common factored denominator to \( \frac{x^2 + x + 3}{6x - 4} \) but the numerator doesn't match either.
None of the options correspond with \( \frac{x^2 - 22x + 3}{3x - 2} \).
It appears either there’s been an inconsistency or misinterpretation of the expressions given. The correct approach yields:
\[
\frac{x^2 - 22x + 3}{3x - 2}
\]
Thus, it remains unclear which specific response matches the correct addition of the expressions, as listed responses do not reflect the accurate calculation of the sum provided from the original problem.
Now we can combine the numerators:
\[
\frac{x^2 - 23x - 2 + x + 5}{3x - 2}
\]
Next, we simplify the numerator:
1. Combine like terms in the numerator:
\[
x^2 - 23x + x - 2 + 5 = x^2 - 22x + 3
\]
Now we have:
\[
\frac{x^2 - 22x + 3}{3x - 2}
\]
None of the provided answer options match our result exactly, but let's look at the closest format and determine which could potentially be correct:
1. **Option 1**: \( \frac{x^2 + x + 3}{3x - 2} \)
2. **Option 2**: \( \frac{x^2 + x + 3}{6x - 4} \)
3. **Option 3**: \( \frac{x^3 + 3}{3x - 2} \)
4. **Option 4**: \( \frac{x^2 + x + 7}{3x - 2} \)
Given our numerator is \(x^2 - 22x + 3\), the first and the fourth options do not fit, and the third option presents a cubic expression, which is not our case.
The second option simplifies with a common factored denominator to \( \frac{x^2 + x + 3}{6x - 4} \) but the numerator doesn't match either.
None of the options correspond with \( \frac{x^2 - 22x + 3}{3x - 2} \).
It appears either there’s been an inconsistency or misinterpretation of the expressions given. The correct approach yields:
\[
\frac{x^2 - 22x + 3}{3x - 2}
\]
Thus, it remains unclear which specific response matches the correct addition of the expressions, as listed responses do not reflect the accurate calculation of the sum provided from the original problem.