Let `P(x) = x^(7) - 3x^(5) +x^(3) - 7x^(2) +5` and `q(x) = x- 2`. The remainder if `p(x)` is divided by `q(x)` is

Options

A. `(5)/(2)`
B. 5
C. 17
D. `-3`

Answer

To find the remainder when the polynomial \( P(x) = x^7 - 3x^5 + x^3 - 7x^2 + 5 \) is divided by \( q(x) = x - 2 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - a \) is equal to \( f(a) \).

### Step-by-step Solution:

1. **Identify the polynomial and the divisor**:
- We have \( P(x) = x^7 - 3x^5 + x^3 - 7x^2 + 5 \).
- The divisor is \( q(x) = x - 2 \), which means \( a = 2 \).

2. **Apply the Remainder Theorem**:
- According to the theorem, the remainder \( R \) when \( P(x) \) is divided by \( q(x) \) is given by:
\[ R = P(2) \]

3. **Calculate \( P(2) \)**:
- Substitute \( x = 2 \) into \( P(x) \):
\[ P(2) = 2^7 - 3(2^5) + 2^3 - 7(2^2) + 5 \]

4. **Evaluate each term**:
- Calculate \( 2^7 = 128 \)
- Calculate \( 3(2^5) = 3 \times 32 = 96 \)
- Calculate \( 2^3 = 8 \)
- Calculate \( 7(2^2) = 7 \times 4 = 28 \)

5. **Substitute the calculated values**:
\[ P(2) = 128 - 96 + 8 - 28 + 5 \]

6. **Perform the arithmetic**:
- First, calculate \( 128 - 96 = 32 \)
- Then, \( 32 + 8 = 40 \)
- Next, \( 40 - 28 = 12 \)
- Finally, \( 12 + 5 = 17 \)

7. **Conclusion**:
- The remainder when \( P(x) \) is divided by \( q(x) \) is \( 17 \).

### Final Answer:
The remainder is \( \boxed{17} \).