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Given that a + b = 1, simplify the algebraic expression: a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab What is its exact value?

Difficulty: Hard

Correct Answer: 0

Explanation:


Introduction / Context:
This is an identity and constraint-based simplification problem. When a + b is fixed, many expressions can be simplified by rewriting higher powers in terms of symmetric sums like (a + b) and ab. The expression looks complex but collapses to a constant under the condition a + b = 1.


Given Data / Assumptions:

  • a + b = 1
  • Expression: a^4 + b^4 - a^3 - b^3 - 2a^2b^2 + ab
  • a and b are real numbers.


Concept / Approach:
Use symmetric identities: a^2 + b^2 = (a + b)^2 - 2ab a^3 + b^3 = (a + b)^3 - 3ab(a + b) a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 Substitute a + b = 1 and simplify step-by-step to show the entire expression becomes 0.


Step-by-Step Solution:

Let s = a + b = 1 and p = ab. Compute a^2 + b^2 = s^2 - 2p = 1 - 2p. Compute a^3 + b^3 = s^3 - 3ps = 1 - 3p. Compute a^4 + b^4 = (a^2 + b^2)^2 - 2p^2 = (1 - 2p)^2 - 2p^2. So a^4 + b^4 = (1 - 4p + 4p^2) - 2p^2 = 1 - 4p + 2p^2. Now substitute into the expression: E = (1 - 4p + 2p^2) - (1 - 3p) - 2p^2 + p. Simplify: E = 1 - 4p + 2p^2 - 1 + 3p - 2p^2 + p. Combine terms: constants cancel, and (-4p + 3p + p) = 0, and (2p^2 - 2p^2) = 0. Therefore E = 0.


Verification / Alternative check:
Pick a simple pair: a = 1 and b = 0 (satisfies a + b = 1). Then expression becomes 1 + 0 - 1 - 0 - 0 + 0 = 0, confirming the result. Any other valid pair will also produce 0 because the simplification is identity-based.


Why Other Options Are Wrong:

1, 2, or 4 could appear if you forget the -2a^2b^2 term or misuse a^4 + b^4 identity. -1 can appear from sign errors when expanding (1 - 2p)^2.


Common Pitfalls:
Expanding powers without using symmetric identities, or forgetting that a^2b^2 = (ab)^2 = p^2. Also, many mistakes come from not grouping like terms carefully.


Final Answer:
0

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