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2016 ISI Objective Solution Problem 1

Problem

The polynomial \(x^7+x^2+1\) is divisible by

    • (A) \(x^5-x^4+x^2-x+1\)             (B) \(x^5-x^4+x^2+1\)
    • (C)   \(x^5+x^4+x^2+x+1\)          (D)   \(x^5-x^4+x^2+x+1\)

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Understanding the Problem:

The problem is easy to understand right?

STOP

Before scrolling down, your first task is to try the problem YOURSELF.

All the best

We will guide you along a short path to the solution in a step by step approach.

Hint 1: 
Try to factorize \(x^7 + x^2 + 1\).

Hint 2: 
Observe that  \(\omega\) and \(\omega^2 \) are the roots of \(x^7 + x^2 + 1\).

Hint 3:
\(x^7 + x^2 + 1\) =  (\(x^2 + x + 1\)).(\(x^5 - x^4 + x^2 -x + 1\))

A shorter solution or approach can always exist. Think about it. If you find an alternative solution or approach, mention it in the comments. We would love to hear something different from you.

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