When number `x^40 + 31` is divided by `x^4 + 1`, the remainder will be?

Options

A. 30
B. 32
C. 16
D. 48

Answer

To find the remainder when \( x^{40} + 31 \) is divided by \( x^4 + 1 \), we can use polynomial long division or the Remainder Theorem. Here, we will use the Remainder Theorem for simplicity.

### Step-by-Step Solution:

1. **Identify the divisor**: We have \( x^4 + 1 \). We want to find the remainder when dividing \( x^{40} + 31 \) by \( x^4 + 1 \).

2. **Set up the equation**: According to the Remainder Theorem, if we divide a polynomial \( f(x) \) by \( g(x) \), the remainder \( r(x) \) can be found by substituting the roots of \( g(x) \) into \( f(x) \).

3. **Find the roots of the divisor**: The polynomial \( x^4 + 1 = 0 \) has roots where \( x^4 = -1 \). This means \( x = e^{i\frac{\pi}{4}}, e^{i\frac{3\pi}{4}}, e^{i\frac{5\pi}{4}}, e^{i\frac{7\pi}{4}} \).

4. **Express \( x^{40} \)**: Since \( x^4 = -1 \), we can express higher powers of \( x \) in terms of \( x^4 \):
\[ x^{40} = (x^4)^{10} = (-1)^{10} = 1. \]

5. **Substitute back into the polynomial**: Now substitute \( x^{40} \) into the original polynomial:
\[ x^{40} + 31 = 1 + 31 = 32. \]

6. **Conclusion**: The remainder when \( x^{40} + 31 \) is divided by \( x^4 + 1 \) is \( 32 \).

### Final Answer:
The remainder is \( 32 \).