i. \( \quad \) Divide \( x^{3}+7 x^{2} y+7 x y^{2}+y^{3} \) by \( x+y \)

To divide the polynomial \( x^{3}+7 x^{2} y+7 x y^{2}+y^{3} \) by \( x+y \), we can use polynomial long division or synthetic division. Let's use long division here. 1. First, we set up the division. We want to divide \( x^3 + 7x^2y + 7xy^2 + y^3 \) by \( x + y \). 2. Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \) to get \( x^2 \). 3. Multiply \( x^2 \) by \( x + y \) to get \( x^3 + x^2y \). 4. Subtract this result from the original polynomial: \[ (x^3 + 7x^2y + 7xy^2 + y^3) - (x^3 + x^2y) = 6x^2y + 7xy^2 + y^3 \] 5. Now, take the leading term \( 6x^2y \) and divide by \( x \) to get \( 6xy \). 6. Multiply \( 6xy \) by \( x + y \) to get \( 6x^2y + 6xy^2 \). 7. Subtract this from the current polynomial: \[ (6x^2y + 7xy^2 + y^3) - (6x^2y + 6xy^2) = xy^2 + y^3 \] 8. Now divide \( xy^2 \) by \( x \) to get \( y^2 \). 9. Multiply \( y^2 \) by \( x + y \) to get \( xy^2 + y^3 \). 10. Subtract: \[ (xy^2 + y^3) - (xy^2 + y^3) = 0 \] The remainder is 0. Thus, the result of the division is: \[ x^2 + 6xy + y^2 \]