Write the quotient and remainder when we divide :
`(x^(3) + 1)` by (x + 1)

Answer

To find the quotient and remainder when dividing \(x^3 + 1\) by \(x + 1\), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We will divide \(x^3 + 1\) (the dividend) by \(x + 1\) (the divisor).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(x^3\) by the leading term of the divisor \(x\):
\[ \frac{x^3}{x} = x^2 \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply \(x^2\) by the entire divisor \(x + 1\):
\[ x^2 \cdot (x + 1) = x^3 + x^2 \]
Subtract this from the original polynomial:
\[ (x^3 + 1) - (x^3 + x^2) = 1 - x^2 \]

### Step 4: Repeat the process
Now we have \(1 - x^2\). We will divide the leading term \(-x^2\) by \(x\):
\[ \frac{-x^2}{x} = -x \]
This is the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply \(-x\) by the divisor \(x + 1\):
\[ -x \cdot (x + 1) = -x^2 - x \]
Subtract this from \(1 - x^2\):
\[ (1 - x^2) - (-x^2 - x) = 1 + x \]

### Step 6: One more time
Now we have \(1 + x\). Divide the leading term \(x\) by \(x\):
\[ \frac{x}{x} = 1 \]
This is the next term of the quotient.

### Step 7: Multiply and subtract
Multiply \(1\) by the divisor \(x + 1\):
\[ 1 \cdot (x + 1) = x + 1 \]
Subtract this from \(1 + x\):
\[ (1 + x) - (x + 1) = 0 \]

### Conclusion
At this point, we have reached a remainder of \(0\). Therefore, the quotient is:
\[ x^2 - x + 1 \]
And the remainder is:
\[ 0 \]

### Final Answer
- **Quotient:** \(x^2 - x + 1\)
- **Remainder:** \(0\)