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. The Remainder Theorem states that the remainder of a polynomial [tex]\( f(x) \)[/tex] divided by [tex]\( x - c \)[/tex] is [tex]\( f(c) \)[/tex].
In this case, we need to find [tex]\( f(2) \)[/tex].
Here's how you do it step by step:
1. Start with the polynomial:
[tex]\[
f(x) = 2x^4 + x^3 - 8x - 1
\][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] into the polynomial:
[tex]\[
f(2) = 2(2)^4 + (2)^3 - 8(2) - 1
\][/tex]
3. Calculate each term:
- [tex]\( 2(2)^4 = 2 \times 16 = 32 \)[/tex]
- [tex]\( (2)^3 = 8 \)[/tex]
- [tex]\( 8 \times 2 = 16 \)[/tex]
4. Substitute these values back into the expression:
[tex]\[
f(2) = 32 + 8 - 16 - 1
\][/tex]
5. Perform the addition and subtraction:
[tex]\[
f(2) = 40 - 16 - 1 = 23
\][/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 [tex]\( f(2) \)[/tex].
Here's how you do it step by step:
1. Start with the polynomial:
[tex]\[
f(x) = 2x^4 + x^3 - 8x - 1
\][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] into the polynomial:
[tex]\[
f(2) = 2(2)^4 + (2)^3 - 8(2) - 1
\][/tex]
3. Calculate each term:
- [tex]\( 2(2)^4 = 2 \times 16 = 32 \)[/tex]
- [tex]\( (2)^3 = 8 \)[/tex]
- [tex]\( 8 \times 2 = 16 \)[/tex]
4. Substitute these values back into the expression:
[tex]\[
f(2) = 32 + 8 - 16 - 1
\][/tex]
5. Perform the addition and subtraction:
[tex]\[
f(2) = 40 - 16 - 1 = 23
\][/tex]
Therefore, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x - 2 \)[/tex] is [tex]\(\boxed{23}\)[/tex].
