Answer :
To solve the given problem [tex]\( \sqrt{12 - 33} \)[/tex]:
1. We start by evaluating the expression inside the square root:
[tex]\[
12 - 33
\][/tex]
2. Subtract [tex]\( 33 \)[/tex] from [tex]\( 12 \)[/tex]:
[tex]\[
12 - 33 = -21
\][/tex]
3. Now, we need to find the square root of this result. The square root of [tex]\(-21\)[/tex] is a complex number, specifically:
[tex]\[
\sqrt{-21} = \sqrt{21}i
\][/tex]
where [tex]\( i \)[/tex] is the imaginary unit, defined as [tex]\( i = \sqrt{-1} \)[/tex].
Given that the problem asks for the remainder in the context of synthetic division and the results obtained involve complex numbers, and considering the choices provided (which are real numbers), it indicates that the correct interpretation in this context does not involve real division remainders.
Thus, considering the steps and the calculations:
- The expression inside the square root yields [tex]\(-21\)[/tex].
This negative result suggests that none of the choices (A. 6, B. 3, C. 5, D. 4) are numerical remainders and therefore do not apply.
Hence, the correct answer is:
[tex]\[
\boxed{-21}
\][/tex]
1. We start by evaluating the expression inside the square root:
[tex]\[
12 - 33
\][/tex]
2. Subtract [tex]\( 33 \)[/tex] from [tex]\( 12 \)[/tex]:
[tex]\[
12 - 33 = -21
\][/tex]
3. Now, we need to find the square root of this result. The square root of [tex]\(-21\)[/tex] is a complex number, specifically:
[tex]\[
\sqrt{-21} = \sqrt{21}i
\][/tex]
where [tex]\( i \)[/tex] is the imaginary unit, defined as [tex]\( i = \sqrt{-1} \)[/tex].
Given that the problem asks for the remainder in the context of synthetic division and the results obtained involve complex numbers, and considering the choices provided (which are real numbers), it indicates that the correct interpretation in this context does not involve real division remainders.
Thus, considering the steps and the calculations:
- The expression inside the square root yields [tex]\(-21\)[/tex].
This negative result suggests that none of the choices (A. 6, B. 3, C. 5, D. 4) are numerical remainders and therefore do not apply.
Hence, the correct answer is:
[tex]\[
\boxed{-21}
\][/tex]
