# Existence of a polynomial.

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#1
I have little progress while working on this STEP question.

I have to determine whether:

There exists a polynomial P such that |P(x) - cos x| <= 10^-6.

I have tried to write the LHS as a piecewise defined function i.e.

Let f(x) = |P(x) - cos|

Case 1:

P(x)<cos x --> cos x - P(x) <= 10^-6

Case 2:

P(x)>=cos x --> P(x) -cos x<=10^-6.

Other than that I am not sure how to continue on this, as I am not given any other information.

Can someone please provide me with hints on how to continue/ what approach I should take for this problem?
0
3 years ago
#2
Perhaps I'm misunderstanding or there's more to the question (this doesn't seem difficult enough for STEP as is), but surely for any polynomial |P(x)-cos(x)| tends to infinity as x tends to infinity (P unbounded, cos bounded) so there isn't such a polynomial.

Unless you're including power series in which case you have your answer to an even higher accuracy.
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3 years ago
#3
(Original post by ThomH97)
Perhaps I'm misunderstanding or there's more to the question (this doesn't seem difficult enough for STEP as is), but surely for any polynomial |P(x)-cos(x)| tends to infinity as x tends to infinity (P unbounded, cos bounded) so there isn't such a polynomial.

Unless you're including power series in which case you have your answer to an even higher accuracy.
It's one part of 4/5-part question, I believe. But yes, your initial approach is correct: any polynomial is unbounded for large enough x.
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