# analytically calculate the inverse of $4^{th}$ order polynomial [closed]

There was a quiz test: Question: which is the inverse polynomial of $$( X^4 + X + 1)$$ ?

a) $$(X^3 + X + 1)$$

b) $$(X^4 + X + 1)$$

c) $$(X^4 + X^2 + 1)$$

d) $$(X^4 + X^3 + 1)$$

e) the correct answer is missing

and the correct answer was (d).

Could you please show me the steps how to find this inverse function?

question

keys

## closed as unclear what you're asking by Yves Daoust, Dietrich Burde, Lord Shark the Unknown, Shailesh, Martin ArgeramiMay 8 at 3:38

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• Have you tried multiplying the given answer with your polynomial?^^ – Riquelme May 7 at 11:31
• @nauok . Would you mind to write clearly (in your question, not in comments) your definition of the "inverse of a polynomial". If not I am afraid that the downvotes will pour down. – JJacquelin May 7 at 11:51
• @JJacquelin I edited it. is there a way you can suggest? – nauok May 7 at 11:59
• You must have misunderstood (or misreproduced) the homework. The functional inverse of the given polynomial is not a polynomial (in fact, it is not invertible). – Yves Daoust May 7 at 12:02
• What is the exact wording of the quiz? Please note that there is something called reciprocal polynomial (or reflected polynomial) which have nothing to do with inverse of a polynomial... – achille hui May 7 at 12:16

Notice that

$$X^4\left(\left(\frac1X\right)^4+\left(\frac1X\right)+1\right)=X^4+X^3+1$$

and that in GF(2), the additive inverse of

$$X^4+X+1$$ is itself.

• Thank you. so all i had to do was to factor x^4 out? – nauok May 7 at 12:40
• I don't think you understood my answer. It shows that with crazy interpretations, b) or d) hold. – Yves Daoust May 7 at 12:52
• AH. OK. i give up. – nauok May 7 at 13:00

Perhaps there is some mis-understanding. The inverse of the function $$y=x^4+x+1$$ is $$x$$ as a function of $$y$$ and that is a complicated multi-valued $$4$$-cycled algebraic function in terms of $$y^\frac{1}{4}$$. You can get some idea of it's complexity by simply solving for the inverse in Mathematica via the command:

Solve[y==x^4+x+1,x]

Also, we can numerically invert it by (numerically) solving the associated monodromy differential equation. To get some idea of what that involves see, Algebraic Functions