If the polynomial $x^{2}+x+1$ is divided by $(x-i)$ over the complex field, then the remainder is:

Q: If the polynomial $x^{2}+x+1$ is divided by $(x-i)$ over the complex field, then the remainder is:

To determine the remainder when the polynomial $ f(x) = x^2 + x + 1 $ is divided by $ x - i $, we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial $ f(x) $ by a linear divisor $ x - c $ is $ f(c) $. In this case, the divisor is $ x - i $, so we need to evaluate the polynomial at $ x = i $: \[ f(i) = i^2 + i + 1. \] Now, let's calculate each term: 1. $ i^2 = -1 $ (since $ i $ is the imaginary unit and $ i^2 = -1 $). 2. Therefore, $ f(i) = -1 + i + 1 $. Simplifying this expression: \[ f(i) = (-1 + 1) + i = 0 + i = i. \] Thus, the remainder when $ x^2 + x + 1 $ is divided by $ x - i $ is $ i $. Now, let's explain why the other options are incorrect: 1. **Option 1: 0** - This would imply that $ x - i $ is a factor of $ x^2 + x + 1 $, which is not the case since the remainder is $ i $, not 0. 2. **Option 3: -i** - This option suggests that the remainder is $-i$, but our calculation shows that the remainder is actually $ i $. 3. **Option 4: $ x + i $** - This option implies a polynomial remainder, which is not possible when dividing by a linear term. The remainder must be a constant, and our calculation shows it is $ i $. Therefore, the correct answer is indeed Option 2, $ i $, as supported by the Remainder Theorem and our calculations.

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To determine the remainder when the polynomial $ f(x) = x^2 + x + 1 $ is divided by $ x - i $, we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial $ f(x) $ by a linear divisor $ x - c $ is $ f(c) $.

In this case, the divisor is $ x - i $, so we need to evaluate the polynomial at $ x = i $:

$$ f(i) = i^2 + i + 1. $$

Now, let's calculate each term:

1. $ i^2 = -1 $ (since $ i $ is the imaginary unit and $ i^2 = -1 $).
2. Therefore, $ f(i) = -1 + i + 1 $.

Simplifying this expression:

$$ f(i) = (-1 + 1) + i = 0 + i = i. $$

Thus, the remainder when $ x^2 + x + 1 $ is divided by $ x - i $ is $ i $.

Now, let's explain why the other options are incorrect:

1. **Option 1: 0** - This would imply that $ x - i $ is a factor of $ x^2 + x + 1 $, which is not the case since the remainder is $ i $, not 0.

2. **Option 3: -i** - This option suggests that the remainder is $-i$, but our calculation shows that the remainder is actually $ i $.

3. **Option 4: $ x + i $** - This option implies a polynomial remainder, which is not possible when dividing by a linear term. The remainder must be a constant, and our calculation shows it is $ i $.

Therefore, the correct answer is indeed Option 2, $ i $, as supported by the Remainder Theorem and our calculations.