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 the division of a polynomial [tex]\( f(x) \)[/tex] by [tex]\( x - c \)[/tex] is equal to [tex]\( f(c) \)[/tex].
In this case, we need to find the value of [tex]\( f(2) \)[/tex]:
1. Plug in [tex]\( x = 2 \)[/tex] 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]
3. Add these results together:
[tex]\[
f(2) = 32 + 8 - 16 - 1
\][/tex]
4. Simplify:
[tex]\[
f(2) = 32 + 8 = 40
\][/tex]
[tex]\[
40 - 16 = 24
\][/tex]
[tex]\[
24 - 1 = 23
\][/tex]
Therefore, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x - 2 \)[/tex] is [tex]\(\boxed{23}\)[/tex].
In this case, we need to find the value of [tex]\( f(2) \)[/tex]:
1. Plug in [tex]\( x = 2 \)[/tex] 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]
3. Add these results together:
[tex]\[
f(2) = 32 + 8 - 16 - 1
\][/tex]
4. Simplify:
[tex]\[
f(2) = 32 + 8 = 40
\][/tex]
[tex]\[
40 - 16 = 24
\][/tex]
[tex]\[
24 - 1 = 23
\][/tex]
Therefore, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x - 2 \)[/tex] is [tex]\(\boxed{23}\)[/tex].
