F.4 maths

1.
(x - 2)(2x + 3)²
(x - 2)(x + 2)²
(x - 2)²(2x + 3)

H.C.F. = x - 2


2.
3x³ - 2x² = x²(3x - 2)
9x³ - 12x² + 4 = x(3x - 2)²

L.C.M. = x²(3x - 2)²


3.
x³ - 5x² + 8x - 4
= x³ - x² - 4x² + 4x + 4x - 4
= (x³ - x²) - (4x² - 4x) + (4x - 4)
= x²(x - 1) - 4x(x - 1) + 4(x - 1)
= (x - 1)(x² - 4x + 4)
= (x - 1)(x - 2)²

x³ + 6x² - 25x + 18
= x³ - x² + 7x² - 7x - 18x + 18
= (x³ - x²) + (7x² - 7x) - (18x - 18)
= x²(x - 1) + 7x(x - 1) - 18(x - 1)
= (x - 1)(x² + 7x - 18)
= (x - 1)(x - 2)(x + 9)

H.C.F. = (x - 1)(x - 2)


4.
[2 / (x + 3)] - [1 / (2x - 1)]
= [2(2x - 1) / (x - 3)(2x - 1)] - [(x + 3) / (x + 3)(2x - 1)]
= [2(2x - 1) - (x + 3)] / (x + 3)(2x - 1)
= (4x - 2 - x - 3) / (x + 3)(2x - 1)
= (3x - 5) / (x + 3)(2x - 1)


5.
[(x + 7) / (x² - 4x - 5)] + [1 / (x + 1)]
= [(x + 7) / (x + 1)(x - 5)] + [1 / (x + 1)]
= [(x + 7) / (x + 1)(x - 5)] + [(x - 5) / (x + 1)(x - 5)]
= [(x + 7) + (x - 5)] / (x + 1)(x - 5)
= (2x + 2) / (x + 1)(x - 5)
= 2(x + 1) / (x + 1)(x - 5)
= 2 /(x - 5)