Finding the domain of the square root of a polynomial

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SUMMARY

The discussion focuses on determining the domain of the function \(\sqrt{x^{12} - x^9 + x^4 - x + 1}\). The key conclusion is that the domain includes all negative values and positive values greater than 1, while the interval (0, 1) requires further analysis. The polynomial can be simplified to show that it is always non-negative by factoring and analyzing its components, confirming that the expression is non-negative across the specified intervals.

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sadhu
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I was just going through some problems in a maths journal for undergraduate level

i found a sum seeming simple but i am not able to solve it completely

find domain of \sqrt{x^{12} - x^9 + x^4 -x +1}i know domain includes all negative value , all positive value >1 , but i can't get anything about (0,1)

thanks in advance...
 
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For x\in [0,1]\Rightarrow x-1\leq0

Now don't bother with the term x12, it is always positive, and factor the rest terms, i.e.

x^{12}-x^9+x^4-x+1=x^{12}-x^4(x^5-1)-(x-1)=x^{12}+x^4(1-x^5)+(1-x)\geq0

as sum of non negative numbers.
 
thanks for that...
i was trying to do with method of intervals
 

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