Divide:
𝑥
2
+
𝑥
−
6
𝑥
−
2

Question

Divide: 
𝑥
2
+
𝑥
−
6
𝑥
−
2

BlackTom AI · Accepted Answer

Consider the division problem: (x^2 + x − 6) ÷ (x − 2).
First, factor the quadratic in the numerator: x^2 + x − 6 = (x + 3)(x − 2).
Now the expression becomes [(x + 3)(x − 2)] ÷ (x − 2).
Since (x − 2) appears in both numerator and denominator, we can cancel the common factor (x − 2), leaving x + 3, with the caveat that x ≠ 2 because division by zero is undefined.
Option by option analysis:
- Option 1: x − 3. This would imply a factorization or cancellation that leads to a difference of signs inconsistent with the actual factorization (x + 3)(x − 2). Therefore this is incorrect.
- Option 2: x + 3. This matches the simplified form after canceling the common factor, so this is the correct result provided x ≠ 2.
- Option 3: x − 2. This would suggest canceling yields 1 times (x − 2) or leaving a factor that did not cancel, which is not what happens after proper factoring and cancellation.
- Option 4: x + 2. This option does not arise from the factorization (x + 3)(x − 2) and is inconsistent with the cancellation steps above.
In summary, the proper simplification is x + 3 (with the restriction x ≠ 2), and the other options do not align with the actual factorization and cancellation process.

Divide: 𝑥 2 + 𝑥 − 6 𝑥 − 2

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