Difficulty: Hard
Correct Answer: 6√6
Explanation:
Introduction / Context:
This question tests advanced simplification with surds (square roots). Rather than expanding x^4 directly (which becomes messy), the standard approach is to exploit structured cancellation: many contest-style polynomials are designed so that substituting x = 1 + √2 + √3 reduces to a simple surd after combining terms. The clean result indicates the expression was crafted for simplification.
Given Data / Assumptions:
Concept / Approach:
A practical method is to compute x^2 and x^3 in simplified surd form, track like terms (constants, √2, √3, √6), and then observe cancellation when substituted into the polynomial. Because √2 * √3 = √6, the basis {1, √2, √3, √6} is enough to represent all intermediate results.
Step-by-Step Solution:
Let x = 1 + √2 + √3
Compute x^2 = (1 + √2 + √3)^2 = 1 + 2 + 3 + 2√2 + 2√3 + 2√6
So x^2 = 6 + 2√2 + 2√3 + 2√6
Use x^3 = x * x^2 and x^4 = (x^2)^2, but keep everything in 1, √2, √3, √6 form
When substituting into 2x^4 - 8x^3 - 5x^2 + 26x - 28, the constant, √2, and √3 parts cancel by design
The remaining non-cancelling term is the √6 component, which simplifies to 6√6
Verification / Alternative check:
A quick numerical check supports the exact form: x ≈ 1 + 1.414 + 1.732 ≈ 4.146. Evaluating the polynomial numerically gives approximately 14.697, and 6√6 ≈ 6*2.449 ≈ 14.697, which matches closely, confirming the exact result.
Why Other Options Are Wrong:
2√2, 4√2, and 3√3 imply the expression collapses to a single √2 or √3 term, but the structure of x includes both √2 and √3, producing √6 terms naturally.
5√5 introduces √5, which cannot appear from combining √2 and √3 (since √2*√3 = √6, not √5).
Common Pitfalls:
Expanding x^4 directly and losing track of terms, or incorrectly multiplying surds (for example treating √2*√3 as √5). Also, dropping cross terms like 2√6 in x^2 leads to wrong final simplification.
Final Answer:
2x^4 - 8x^3 - 5x^2 + 26x - 28 = 6√6
Discussion & Comments