What is the result when the fraction
𝑡
𝑡
+
2
−
1
+
𝑡
𝑡
−
3
is simplified as far as possible?

Question

What is the result when the fraction 
𝑡
𝑡
+
2
−
1
+
𝑡
𝑡
−
3
 is simplified as far as possible?

BlackTom AI · Accepted Answer

Assuming the expression is t/(t+2) − 1 + t/(t−3), use a common denominator (t+2)(t−3) to combine the three terms.

1) Knowledge points
- To add/subtract rational expressions, rewrite each term with the same denominator.
- Convert integer 1 to a rational expression with that denominator.
- Expand and combine like terms in the numerator, and simplify any common factors.

2) Step-by-step combination
Common denominator: (t+2)(t−3).

Rewrite each term:
- t/(t+2) = t(t−3)/[(t+2)(t−3)] because multiply numerator and denominator by (t−3).
- 1 = (t+2)(t−3)/[(t+2)(t−3)].
- t/(t−3) = t(t+2)/[(t+2)(t−3)] because multiply numerator and denominator by (t+2).

Add/subtract numerators:
Numerator = t(t−3) − (t+2)(t−3) + t(t+2).

3) Intermediate expansions
- t(t−3) = t^2 − 3t
- (t+2)(t−3) = t^2 − t − 6, so −(t+2)(t−3) = −t^2 + t + 6
- t(t+2) = t^2 + 2t

Now sum:
(t^2 − 3t) + (−t^2 + t + 6) + (t^2 + 2t)
= (t^2 − t^2 + t^2) + (−3t + t + 2t) + 6
= t^2 + 0t + 6
= t^2 + 6

Therefore the combined fraction is (t^2 + 6)/[(t+2)(t−3)].

4) Why this is correct
Each term was converted to the common denominator and arithmetic done term-by-term. The cancellations in the linear terms (−3t + t + 2t = 0) are algebraically correct, leaving t^2 + 6 in the numerator and (t+2)(t−3) in the denominator. The numerator does not factor over the integers, so the fraction is already as simple as possible (except to expand the denominator if desired: t^2 − t − 6).

5) Other possible interpretations / mismatch with the provided answer
The provided answer in your file appears as a polynomial string (−6t − 2t^2 − t − 6) which, when combined, equals −2t^2 − 7t − 6. That does not match the correct simplified rational expression derived above, (t^2 + 6)/(t^2 − t − 6). Either the original expression was formatted differently (for example a complex fraction or a different arrangement of signs), or the provided answer string was corrupted. If the intended expression is different, please paste it exactly (using parentheses as needed). If the intended expression was t/(t+2) − 1 + t/(t−3), the correct simplified form is (t^2 + 6)/(t^2 − t − 6), not the polynomial shown.

Please confirm the exact original expression or the desired format so I can produce an explanation that matches the provided answer (−6t − 2t^2 − t − 6) if that is indeed correct.

What is the result when the fraction 𝑡 𝑡 + 2 − 1 + 𝑡 𝑡 − 3 is simplified as far as possible?Single choice

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