Answer :

To find the remainder when [tex]\( f(x) = 2x^4 + x^3 - 8x - 1 \)[/tex] is divided by [tex]\( x-2 \)[/tex], we can use the Remainder Theorem. This theorem states that the remainder of the division of a polynomial [tex]\( f(x) \)[/tex] by a linear divisor [tex]\( x-a \)[/tex] is simply [tex]\( f(a) \)[/tex].

Here's how you can apply the Remainder Theorem to solve this problem step-by-step:

1. Identify the Value of [tex]\( a \)[/tex]:
The divisor given is [tex]\( x-2 \)[/tex], which means [tex]\( a = 2 \)[/tex].

2. Substitute [tex]\( x = a \)[/tex] into [tex]\( f(x) \)[/tex]:
We need to evaluate [tex]\( f(2) \)[/tex] to find the remainder.

Here's the calculation:
[tex]\[
f(2) = 2(2)^4 + (2)^3 - 8(2) - 1
\][/tex]

3. Calculate Each Term:
- Calculate [tex]\( 2(2)^4 \)[/tex]:
[tex]\[
2 \times 16 = 32
\][/tex]

- Calculate [tex]\( (2)^3 \)[/tex]:
[tex]\[
2 \times 2 \times 2 = 8
\][/tex]

- Calculate [tex]\(-8(2)\)[/tex]:
[tex]\[
-8 \times 2 = -16
\][/tex]

- The constant [tex]\(-1\)[/tex] remains the same.

4. Add All the Terms Together:
Combine all these results:
[tex]\[
32 + 8 - 16 - 1 = 23
\][/tex]

5. Conclusion:
The remainder when [tex]\( f(x) = 2x^4 + x^3 - 8x - 1 \)[/tex] is divided by [tex]\( x-2 \)[/tex] is [tex]\( 23 \)[/tex].

So, the correct answer is [tex]\(\boxed{23}\)[/tex].

Thanks for taking the time to read What is the remainder when tex f x 2x 4 x 3 8x 1 tex is divided by tex x 2 tex A 23 B. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada