Answer :
To solve for the remainder when [tex]\( f(x) = 2x^4 + x^3 - 8x - 1 \)[/tex] is divided by [tex]\( x - 2 \)[/tex], we can apply 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 [tex]\( f(c) \)[/tex].
For this problem, [tex]\( c \)[/tex] is 2 because we are dividing by [tex]\( x - 2 \)[/tex]. Thus, we need to evaluate [tex]\( f(2) \)[/tex].
Let's substitute [tex]\( x = 2 \)[/tex] into the polynomial and calculate:
[tex]\[
f(2) = 2(2)^4 + (2)^3 - 8(2) - 1
\][/tex]
Calculate each term:
1. [tex]\( 2(2)^4 = 2 \times 16 = 32 \)[/tex]
2. [tex]\( (2)^3 = 8 \)[/tex]
3. [tex]\( -8(2) = -16 \)[/tex]
Now, substitute these values back into the expression:
[tex]\[
f(2) = 32 + 8 - 16 - 1
\][/tex]
Perform the arithmetic step-by-step:
- Calculate [tex]\( 32 + 8 = 40 \)[/tex]
- Calculate [tex]\( 40 - 16 = 24 \)[/tex]
- Finally, calculate [tex]\( 24 - 1 = 23 \)[/tex]
Thus, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x - 2 \)[/tex] is 23.
So, the correct answer is:
B. 23
For this problem, [tex]\( c \)[/tex] is 2 because we are dividing by [tex]\( x - 2 \)[/tex]. Thus, we need to evaluate [tex]\( f(2) \)[/tex].
Let's substitute [tex]\( x = 2 \)[/tex] into the polynomial and calculate:
[tex]\[
f(2) = 2(2)^4 + (2)^3 - 8(2) - 1
\][/tex]
Calculate each term:
1. [tex]\( 2(2)^4 = 2 \times 16 = 32 \)[/tex]
2. [tex]\( (2)^3 = 8 \)[/tex]
3. [tex]\( -8(2) = -16 \)[/tex]
Now, substitute these values back into the expression:
[tex]\[
f(2) = 32 + 8 - 16 - 1
\][/tex]
Perform the arithmetic step-by-step:
- Calculate [tex]\( 32 + 8 = 40 \)[/tex]
- Calculate [tex]\( 40 - 16 = 24 \)[/tex]
- Finally, calculate [tex]\( 24 - 1 = 23 \)[/tex]
Thus, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x - 2 \)[/tex] is 23.
So, the correct answer is:
B. 23
