Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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
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
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.