4 a Divide \( x^{4}-1 \) by \( (x+1) \)

To divide \( x^4 - 1 \) by \( x + 1 \), we can use polynomial long division. 1. **Set up the division**: We want to divide \( x^4 - 1 \) by \( x + 1 \). 2. **Divide the leading term**: The leading term of \( x^4 \) divided by the leading term of \( x + 1 \) gives us \( x^3 \). 3. **Multiply and subtract**: Multiply \( x^3 \) by \( x + 1 \): \[ x^3(x + 1) = x^4 + x^3 \] Now subtract this from \( x^4 - 1 \): \[ (x^4 - 1) - (x^4 + x^3) = -x^3 - 1 \] 4. **Repeat the process**: Now divide the leading term \(-x^3\) by \(x\): \[ -x^3 \div x = -x^2 \] Multiply \(-x^2\) by \(x + 1\): \[ -x^2(x + 1) = -x^3 - x^2 \] Subtract this from \(-x^3 - 1\): \[ (-x^3 - 1) - (-x^3 - x^2) = x^2 - 1 \] 5. **Continue dividing**: Now divide \(x^2\) by \(x\): \[ x^2 \div x = x \] Multiply \(x\) by \(x + 1\): \[ x(x + 1) = x^2 + x \] Subtract this from \(x^2 - 1\): \[ (x^2 - 1) - (x^2 + x) = -x - 1 \] 6. **Final division**: Now divide \(-x\) by \(x\): \[ -x \div x = -1 \] Multiply \(-1\) by \(x + 1\): \[ -1(x + 1) = -x - 1 \] Subtract this from \(-x - 1\): \[ (-x - 1) - (-x - 1) = 0 \] Since the remainder is 0, we have completed the division. The result of dividing \( x^4 - 1 \) by \( x + 1 \) is: \[ \boxed{x^3 - x^2 + x - 1} \]