Finding the domain of the square root of a polynomial

In summary, the conversation was about a problem in a maths journal for undergraduate level that seemed simple but the person was having trouble solving it. The problem was to find the domain of the expression sqrt(x^12 - x^9 + x^4 - x + 1), and the person was able to determine that the domain includes all negative values and all positive values greater than 1, but was unsure about values between 0 and 1. They were given a helpful tip to factor the expression and use the method of intervals to solve it.
  • #1
sadhu
157
0
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 [tex]\sqrt{x^{12} - x^9 + x^4 -x +1}[/tex]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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  • #2
For [tex]x\in [0,1]\Rightarrow x-1\leq0[/tex]

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

[tex]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[/tex]

as sum of non negative numbers.
 
  • #3
thanks for that...
i was trying to do with method of intervals
 

What is the domain of the square root of a polynomial?

The domain of the square root of a polynomial is all real numbers that make the expression under the radical (inside the square root) greater than or equal to zero.

How do I find the domain of the square root of a polynomial?

To find the domain of the square root of a polynomial, set the expression under the radical greater than or equal to zero and solve for the variable. The resulting values will be the domain.

Why is it important to find the domain of the square root of a polynomial?

It is important to find the domain of the square root of a polynomial to ensure that the expression is defined for all possible values of the variable. This helps to avoid dividing by zero or taking the square root of a negative number, which are undefined operations.

Are there any restrictions on the domain of the square root of a polynomial?

Yes, there are a few restrictions on the domain of the square root of a polynomial. The expression under the radical cannot be negative, so any values that make it negative must be excluded from the domain. Additionally, if the polynomial contains a variable in the denominator, the variable cannot take on values that would make the denominator equal to zero.

What happens if I include values in the domain that are not allowed?

If values are included in the domain that are not allowed, the expression will be undefined at those points and will not have a real solution. It is important to always check for restrictions on the domain to ensure that the expression is defined for all possible values of the variable.

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