Evaluate the expression when x = 1 + √2: compute x^4 - 4x^3 + 4x^2 (give a numeric value).

Difficulty: Medium

-1
0
1
2

Correct Answer: 1

Explanation:

Introduction / Context:
This problem checks algebraic manipulation and recognition of patterns. Substituting x = 1 + √2 into a structured polynomial often simplifies using identities.

Given Data / Assumptions:

x = 1 + √2.
Compute S = x^4 - 4x^3 + 4x^2.

Concept / Approach:
Observe that x^4 - 4x^3 + 4x^2 = x^2(x^2 - 4x + 4) = x^2 (x - 2)^2. This factorization greatly simplifies evaluation.

Step-by-Step Solution:

S = x^2 (x - 2)^2x - 2 = (1 + √2) - 2 = √2 - 1x^2 = (1 + √2)^2 = 1 + 2√2 + 2 = 3 + 2√2(x - 2)^2 = (√2 - 1)^2 = 2 - 2√2 + 1 = 3 - 2√2S = (3 + 2√2)(3 - 2√2) = 9 - (2√2)^2 = 9 - 8 = 1

Verification / Alternative check:
Numeric approximation: √2 ≈ 1.4142, x ≈ 2.4142; computing S directly gives ~1, confirming the exact result.

Why Other Options Are Wrong:
-1, 0, and 2 arise from arithmetic mistakes or failing to factor and simplify correctly.

Common Pitfalls:
Expanding without noticing the perfect-square structure; sign errors when squaring (√2 - 1); rounding too early during approximations.

Evaluate the expression when x = 1 + √2: compute x^4 - 4x^3 + 4x^2 (give a numeric value).

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