`x^(35) +3` is divided by `x^(5)+1`, find remainder

Answer

To find the remainder when \( x^{35} + 3 \) is divided by \( x^5 + 1 \), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = x^{35} + 3 \). We want to find the remainder when \( f(x) \) is divided by \( x^5 + 1 \).

### Step 2: Set up the equation
Since we are dividing by \( x^5 + 1 \), we can set \( x^5 + 1 = 0 \). This gives us \( x^5 = -1 \).

### Step 3: Substitute \( x^5 \) into the function
We can express \( x^{35} \) in terms of \( x^5 \):
\[ x^{35} = (x^5)^7 \]
Substituting \( x^5 = -1 \) into the equation:
\[ x^{35} = (-1)^7 = -1 \]

### Step 4: Substitute back into the function
Now substitute \( x^{35} \) back into the function \( f(x) \):
\[ f(x) = x^{35} + 3 = -1 + 3 \]

### Step 5: Calculate the value
Now, calculate the value:
\[ -1 + 3 = 2 \]

### Conclusion
Thus, the remainder when \( x^{35} + 3 \) is divided by \( x^5 + 1 \) is \( 2 \).

### Summary
The remainder is \( \boxed{2} \).

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