1.
Simplify: (3x² + yz)(3x² - yz).
Correct answer: 9x⁴ - y²z². This is a difference of squares: (3x² + yz)(3x² - yz) = (3x²)² - (yz)² = 9x⁴ - y²z².
2.
Simplify: (x + 1)(x - 3)(x - 1).
Correct answer: x³ - 3x² - x + 3. First multiply (x + 1)(x - 3) = x² - 2x - 3. Then (x² - 2x - 3)(x - 1) = x³ - x² - 2x² + 2x - 3x + 3 = x³ - 3x² - x + 3.
3.
Multiply: (3a² + b³)(3a² - b³).
Correct answer: 9a⁴ - b⁶. This is a difference of squares: (3a² + b³)(3a² - b³) = (3a²)² - (b³)² = 9a⁴ - b⁶.
4.
Simplify: (3x - 1)³.
Correct answer: 27x³ - 27x² + 9x - 1. (3x - 1)³ = (3x - 1)(3x - 1)(3x - 1). First, (3x - 1)(3x - 1) = 9x² - 6x + 1. Then (9x² - 6x + 1)(3x - 1) = 27x³ - 27x² + 9x - 1.
5.
Expand and simplify: (10v - 13w)(10v + 13w).
Correct answer: 100v² - 169w². This is a difference of squares: (10v - 13w)(10v + 13w) = (10v)² - (13w)² = 100v² - 169w².
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