Question 1: Simplify the expression: x^2 - 16.

Answer: (x - 4)(x + 4)

Question 2: Factor the expression: 25y^2 - 9.

Answer: (5y - 3)(5y + 3)

Question 3: Identify and correct the mistake: Factor x^2 + 36 as a difference of squares.

Answer: Incorrect because x^2 + 36 cannot be written as a difference of squares; it's a sum of squares.

Question 4: Express the difference of squares for 49 - y^2.

Answer: (7 - y)(7 + y)

Question 5: Plot the graph of y = x^2 - 9 on the grid. What type of pattern do you observe?

Answer: A parabola opening upwards with x-intercepts at x = -3 and x = 3.

Question 6: Factor the expression: 81z^2 - 16.

Answer: (9z - 4)(9z + 4)

Question 7: A student attempts to factor y^2 + 36. Explain the misconception and correct it.

Answer: Misconception: y^2 + 36 is factored as (y + 6)(y - 6). Correct: sum of squares cannot be factored over the real numbers.

Question 8: Identify the error: Factor x^2 - 25 as a difference of squares.

Answer: The error is assuming x^2 - 25 can be factored as (x - 5)(x + 5) without recognizing that it's correct; this is actually the proper factorization.

Question 9: Find two numbers that multiply to 36 and add to 13. Use these to factor x^2 + 13x + 36.

Answer: (x + 4)(x + 9)

Question 10: Construct an algebraic expression that illustrates the difference of squares involving a variable y.

Answer: y^2 - 49

Question 11: Explain why x^2 + 4 cannot be factored as a difference of squares.

Answer: Because it is a sum of squares, not a difference, and sum of squares cannot be factored over real numbers as a difference of two squares.