Answer :

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

Here, we are dividing by [tex]\( x-2 \)[/tex], which means [tex]\( a = 2 \)[/tex]. So, we need to find [tex]\( f(2) \)[/tex].

Let's evaluate [tex]\( f(2) \)[/tex]:

1. Substitute 2 into the polynomial:

[tex]\( f(2) = 2(2)^4 + (2)^3 - 8(2) - 1 \)[/tex].

2. Calculate each term:

- [tex]\( 2(2)^4 = 2 \times 16 = 32 \)[/tex],
- [tex]\( (2)^3 = 8 \)[/tex],
- [tex]\( -8(2) = -16 \)[/tex],
- The constant term is [tex]\(-1\)[/tex].

3. Add them together:

[tex]\( f(2) = 32 + 8 - 16 - 1 \)[/tex].

4. Simplify:

- [tex]\( 32 + 8 = 40 \)[/tex],
- [tex]\( 40 - 16 = 24 \)[/tex],
- [tex]\( 24 - 1 = 23 \)[/tex].

Thus, the remainder is 23. So, the answer is:

B. 23

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