To find the remainder when the polynomial [tex]\(x^{11} + 101\)[/tex] is divided by [tex]\(x + 1\)[/tex], we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial [tex]\(f(x)\)[/tex] by a linear divisor [tex]\(x - a\)[/tex] is given by [tex]\(f(a)\)[/tex].
In our case, we have the polynomial [tex]\(f(x) = x^{11} + 101\)[/tex] and the divisor [tex]\(x + 1\)[/tex], which we can rewrite as [tex]\(x - (-1)\)[/tex]. Here, [tex]\(a = -1\)[/tex].
According to the Remainder Theorem:
[tex]\[ \text{Remainder} = f(-1) \][/tex]
We need to evaluate the polynomial [tex]\(f(x) = x^{11} + 101\)[/tex] at [tex]\(x = -1\)[/tex]:
[tex]\[ f(-1) = (-1)^{11} + 101 \][/tex]
Step-by-step calculation:
1. Calculate [tex]\((-1)^{11}\)[/tex]:
[tex]\[ (-1)^{11} = -1 \][/tex]
2. Add 101 to [tex]\(-1\)[/tex]:
[tex]\[ f(-1) = -1 + 101 = 100 \][/tex]
Therefore, the remainder when [tex]\(x^{11} + 101\)[/tex] is divided by [tex]\(x + 1\)[/tex] is [tex]\(100\)[/tex].
The correct answer is:
c) 100
