To analyze the assertion and reason provided, let’s first rewrite the assertion and then look into the reason.
Assertion: The equation \( a^3 + b^3 + 3ab - 1 = (a + b - 1)(a^2 + b^2 + a + b - ab + 1) \) needs to be verified for correctness.
Reason: The identity \( a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 + ab + ac + bc) \) is stated, presumably to help prove the assertion.
Verification of Assertion:
Let's simplify the right-hand side of the assertion.
- Expanding the right-hand side:
\[
(a + b - 1)(a^2 + b^2 + a + b - ab + 1)
\]
We expand this step-by-step:
- \( (a + b)(a^2 + b^2 + a + b - ab + 1) \)
- \( -1(a^2 + b^2 + a + b - ab + 1) \)
Expanding \( (a + b)(a^2 + b^2 + a + b - ab + 1) \) gives: \[ a(a^2 + b^2 + a + b - ab + 1) + b(a^2 + b^2 + a + b - ab + 1) \] Which simplifies to: \[ a^3 + ab^2 + a^2 + ab + a - a^2b + a
- b^3 + ba^2 + b^2 + ab + b - ab^2 + b \]
Collecting like terms gives: \[ a^3 + b^3 + a^2 + b^2 + 2ab + a + b - a^2b - ab^2 \]
Now, consider the other term: \[ -1(a^2 + b^2 + a + b - ab + 1) \] This yields: \[ -a^2 - b^2 - a - b + ab - 1 \]
Putting it all together: \[ (a^3 + b^3 + 2ab + a + b - a^2b - ab^2) + (-a^2 - b^2 - a - b + ab - 1) \]
Simplification:
The corresponding terms reduce: \[ a^3 + b^3 + ab + (2ab - ab) - (a^2 + b^2 + a + b) - 1 \]
Collecting all gives \( a^3 + b^3 + 3ab - 1 \), which validates the assertion.
Final Check with Reason:
The reasoning presented is not directly related to the assertion, as it involves a different identity. Thus, while the assertion is true, the reason presented does not logically support or explain that assertion directly.
Conclusion:
Assertion: True
Reason: Not directly relevant or supportive for the assertion.