F.2 數學題

1a)x² - 26xy - 27y²
= (x² - 27xy) + (xy - 27y²)
= x(x - 27y) + y(x - 27y)
= (x + y) (x - 27y)
b)x⁶ - 26x³y³ - 27y⁶
= (x³)² - 26(x³)(y³) - 27(y³)²By a) ,
= (x³ + y³) (x³ - 27y³)
= (x³ + y³) (x³ - (3y)³)
= (x + y)(x² - xy + y²) (x - 3y)(x² + x(3y) + (3y)²)
= (x + y) (x² - xy + y²) (x - 3y) (x² + 3xy + 9y²)
2a)x³ - 8
= x³ - 2³
= (x - 2) (x² + x(2) + 2²)
= (x - 2) (x² + 2x + 4)
b)x³ - 6x + 4
= (x³ - 8) - 6x + 12
= (x - 2) (x² + 2x + 4) - (6x - 12)
= (x - 2) (x² + 2x + 4) - 6 (x - 2)
= (x - 2) (x² + 2x + 4 - 6)
= (x - 2) (x² + 2x - 2)
3) (x - 1)(x - 3)(x - 5)(x - 7) - 65
= (x - 1)(x - 7) (x - 3)(x - 5) - 65
= (x² - 8x + 7) (x² - 8x + 15) - 65
= (x² - 8x + 11 - 4) (x² - 8x + 11 + 4) - 65
= (x² - 8x + 11)² - 4² - 65
= (x² - 8x + 11)² - 81
= (x² - 8x + 11 - 9) (x² - 8x + 11 + 9)
= (x² - 8x + 2) (x² - 8x + 20)