Which is a valid factorization of x^2 - 9?

Which is a valid factorization of x^2 - 9?

• A. (x-9)(x+1)
• B. (x-3)(x+9)
• C. (x-3)(x+3)
• D. (x+3)^2

Answer: (C)

Recognize a difference of squares. An expression like x^2 - 9 is a^2 - b^2 with a = x and b = 3. It factors as (x - 3)(x + 3) because multiplying those gives x^2 + 3x - 3x - 9, which simplifies to x^2 - 9. This is the valid factorization. The other products would introduce extra x terms or constants that don’t cancel to produce x^2 - 9, for example: (x - 9)(x + 1) expands to x^2 - 8x - 9; (x - 3)(x + 9) expands to x^2 + 6x - 27; (x + 3)^2 expands to x^2 + 6x + 9.

Recognize a difference of squares. An expression like x^2 - 9 is a^2 - b^2 with a = x and b = 3. It factors as (x - 3)(x + 3) because multiplying those gives x^2 + 3x - 3x - 9, which simplifies to x^2 - 9. This is the valid factorization. The other products would introduce extra x terms or constants that don’t cancel to produce x^2 - 9, for example: (x - 9)(x + 1) expands to x^2 - 8x - 9; (x - 3)(x + 9) expands to x^2 + 6x - 27; (x + 3)^2 expands to x^2 + 6x + 9.