When `x^(40)+2` is divided by `x^(4)+1`, then what is the remainder?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the remainder when \( x^{40} + 2 \) is divided by \( x^4 + 1 \), we can use the property of polynomial division. The remainder will be a polynomial of degree less than the divisor, which in this case is \( x^4 + 1 \). Therefore, the remainder can be expressed in the form \( ax^3 + bx^2 + cx + d \).

### Step 1: Identify the relationship
We know that if \( x^4 + 1 = 0 \), then \( x^4 = -1 \). This will help us reduce higher powers of \( x \).

**Hint:** Substitute \( x^4 \) with \(-1\) to simplify higher powers of \( x \).

### Step 2: Simplify \( x^{40} \)
We can express \( x^{40} \) in terms of \( x^4 \):
\[ x^{40} = (x^4)^{10} = (-1)^{10} = 1 \]

**Hint:** Use the property of exponents to rewrite \( x^{40} \).

### Step 3: Substitute back into the expression
Now we substitute \( x^{40} \) back into the original expression:
\[ x^{40} + 2 = 1 + 2 = 3 \]

**Hint:** Combine the simplified term with the constant to find the final value.

### Step 4: Conclusion
Since \( x^{40} + 2 \) simplifies to \( 3 \), this means that when \( x^{40} + 2 \) is divided by \( x^4 + 1 \), the remainder is simply \( 3 \).

### Final Answer
The remainder when \( x^{40} + 2 \) is divided by \( x^4 + 1 \) is \( \boxed{3} \).