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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 simply [tex]\( f(c) \)[/tex].
Let's apply the Remainder Theorem step by step:
1. Identify [tex]\( c \)[/tex] in the divisor [tex]\( x - c \)[/tex]. In this case, since the divisor is [tex]\( x - 2 \)[/tex], we have [tex]\( c = 2 \)[/tex].
2. Substitute [tex]\( x = 2 \)[/tex] into the polynomial [tex]\( f(x) = 2x^4 + x^3 - 8x - 1 \)[/tex].
3. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 2(2)^4 + (2)^3 - 8(2) - 1
\][/tex]
[tex]\[
= 2(16) + 8 - 16 - 1
\][/tex]
[tex]\[
= 32 + 8 - 16 - 1
\][/tex]
[tex]\[
= 40 - 16 - 1
\][/tex]
[tex]\[
= 24 - 1
\][/tex]
[tex]\[
= 23
\][/tex]
Thus, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x - 2 \)[/tex] is 23.
The correct answer is B. 23.
Let's apply the Remainder Theorem step by step:
1. Identify [tex]\( c \)[/tex] in the divisor [tex]\( x - c \)[/tex]. In this case, since the divisor is [tex]\( x - 2 \)[/tex], we have [tex]\( c = 2 \)[/tex].
2. Substitute [tex]\( x = 2 \)[/tex] into the polynomial [tex]\( f(x) = 2x^4 + x^3 - 8x - 1 \)[/tex].
3. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 2(2)^4 + (2)^3 - 8(2) - 1
\][/tex]
[tex]\[
= 2(16) + 8 - 16 - 1
\][/tex]
[tex]\[
= 32 + 8 - 16 - 1
\][/tex]
[tex]\[
= 40 - 16 - 1
\][/tex]
[tex]\[
= 24 - 1
\][/tex]
[tex]\[
= 23
\][/tex]
Thus, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x - 2 \)[/tex] is 23.
The correct answer is B. 23.
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