Question 1: Expand (x + y + z)^3 step-by-step and simplify your expression.

Answer: x^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + y^3 + 3y^2z + 3yz^2 + z^3

Question 2: Calculate the expansion of (2a + 3b + c)^3 using the step-by-step method.

Answer: 8a^3 + 36a^2b + 24a^2c + 54ab^2 + 36abc + 18ac^2 + 27b^3 + 27b^2c + 9bc^2 + c^3

Question 3: A box contains a cube with side length (x + y + z). Express the volume of the cube as an expanded polynomial.

Answer: x^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + y^3 + 3y^2z + 3yz^2 + z^3

Question 4: Expand (a + b + c)^3 and identify the coefficient of a^2b.

Answer: 3a^2b

Question 5: Using a step-by-step process, expand and simplify (m + n + p)^3.

Answer: m^3 + 3m^2n + 3m^2p + 3mn^2 + 6mnp + 3mp^2 + n^3 + 3n^2p + 3np^2 + p^3

Question 6: A student incorrectly expands (x + y + z)^3 as x^3 + y^3 + z^3. Identify two common mistakes in this approach.

Answer: Ignoring the middle terms; failing to apply the binomial expansion correctly

Question 7: Construct a cubic polynomial by expanding (a + 2b + 3c)^3. Show each step clearly.

Answer: a^3 + 12a^2b + 18a^2c + 24ab^2 + 36abc + 54ac^2 + 8b^3 + 36b^2c + 54bc^2 + 27c^3

Question 8: Solve for x: If (x + y + z)^3 = x^3 + y^3 + z^3 + 3xy(x + y) + 3xz(x + z), what is the expansion of the left side?

Answer: x^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + y^3 + 3y^2z + 3yz^2 + z^3

Question 9: Expand (a + b + c)^3 and verify the coefficient of the term a^2c.

Answer: 3a^2c

Question 10: Challenge: Write a step-by-step expansion for (x + y + z)^3 and then factor the resulting polynomial if possible.

Answer: Expanded: x^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + y^3 + 3y^2z + 3yz^2 + z^3; No further factorization into simpler real factors.