Question 1: Expand the expression (x + 2)(x + 3)(x + 4) step by step. Write each stage clearly.

Answer: x^3 + 9x^2 + 26x + 24

Question 2: A student expands (x + 1)(x + 2)(x + 3) and writes the answer as x^3 + 6x^2 + 11x + 6. Identify and explain the mistake made.

Answer: The student incorrectly added coefficients; the correct expansion is x^3 + 6x^2 + 11x + 6. The mistake was in the middle term calculation.

Question 3: Construct the expansion of (x + 3)(x + 4)(x + 5) and compare it with your previous answer. What pattern do you observe?

Answer: The middle coefficient is the sum of the constants, and the constant term is the product of the constants.

Question 4: Expand (x + 2)(x + 3)(x + 4) correctly. Show each step.

Answer: First expand (x + 2)(x + 3) = x^2 + 5x + 6. Then multiply by (x + 4): x^3 + 4x^2 + 5x^2 + 20x + 6x + 24 = x^3 + 9x^2 + 26x + 24.

Question 5: A misconception is to multiply the first two brackets and then multiply the result by the third. Why is this approach incorrect for triple brackets?

Answer: Because it ignores the distributive process across all three factors simultaneously, potentially missing terms or miscalculating coefficients.

Question 6: Expand (x + 1)(x + 2)(x + 3) step-by-step, showing common errors students make and how to avoid them.

Answer: Correct expansion: (x + 1)(x + 2) = x^2 + 3x + 2; then multiply by (x + 3): x^3 + 3x^2 + 2x + 3x^2 + 9x + 6 = x^3 + 6x^2 + 11x + 6.

Question 7: Plot the graph of y = (x + 1)(x + 2)(x + 3) on the grid. What key features do you observe?

Answer: The graph is cubic, with roots at x = -1, -2, -3, and it opens upwards.

Question 8: Expand (x + a)(x + b)(x + c), where a, b, c are constants. Derive a general formula for the expansion.

Answer: x^3 + (a + b + c)x^2 + (ab + ac + bc)x + abc.

Question 9: Given the expansion of (x + 2)(x + 3)(x + 4), identify the coefficients of each term and explain their significance.

Answer: x^3 coefficient: 1, x^2 coefficient: 9, x coefficient: 26, constant: 24. These are derived from the sums and products of the constants.

Question 10: Identify and correct the error in this student solution: expanding (x + 2)(x + 3)(x + 4) to x^3 + 8x^2 + 24x + 24.

Answer: The correct expansion is x^3 + 9x^2 + 26x + 24. The mistake was in calculating the coefficient of x^2 as 8 instead of 9.

Question 11: Challenge: Expand (x + 1)(x + 2)(x + 3) but intentionally make a common mistake. Then, identify and explain the error.

Answer: Incorrect expansion: x^3 + 5x^2 + 6x + 6. Error: Mixing terms during expansion, missing the correct combination of coefficients.