If `4x^(3)-2x^(2)+5x-8` is divided by (x-2), what will be the remainder ?

Options

A. 42
B. 26
C. 81
D. 16

Answer

To find the remainder when the polynomial \( 4x^3 - 2x^2 + 5x - 8 \) is divided by \( x - 2 \), we can use the Remainder Theorem. According to the theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
The polynomial is \( f(x) = 4x^3 - 2x^2 + 5x - 8 \) and the divisor is \( x - 2 \).

2. **Set the divisor equal to zero to find \( c \)**:
\[ x - 2 = 0 \implies x = 2 \]

3. **Substitute \( c = 2 \) into the polynomial**:
We need to calculate \( f(2) \):
\[ f(2) = 4(2)^3 - 2(2)^2 + 5(2) - 8 \]

4. **Calculate each term**:
- Calculate \( 4(2)^3 \):
\[ 4(2^3) = 4(8) = 32 \]
- Calculate \( -2(2)^2 \):
\[ -2(2^2) = -2(4) = -8 \]
- Calculate \( 5(2) \):
\[ 5(2) = 10 \]
- The constant term is \( -8 \).

5. **Combine the results**:
Now, substitute these values back into the equation:
\[ f(2) = 32 - 8 + 10 - 8 \]

6. **Perform the arithmetic**:
- First, combine \( 32 - 8 = 24 \)
- Then, \( 24 + 10 = 34 \)
- Finally, \( 34 - 8 = 26 \)

7. **Conclusion**:
The remainder when \( 4x^3 - 2x^2 + 5x - 8 \) is divided by \( x - 2 \) is \( 26 \).

### Final Answer:
The remainder is \( 26 \).