F4 maths (rational functions)

1) Find the HCF and LCM of x+3, x^2-9 and x^3+27
x+3 , x^2 - 3^2 and x^3 + 3^3
x+3 , (x-3)(x+3) and (x+3)(x^2 - 3x + 9)
HCF = x+3
LCM = (x+3)(x-3)(x^2 - 3x + 9)
2)Simplify (1/4x+1 - 1/1-4x) / (4x^2+2x / 4x^2 + 7x -2)
= [(1-4x) - (4x+1)] / (4x+1)(1-4x)] / (4x^2+2x / 4x^2 + 7x -2)
= [- 8x / (4x+1)(1-4x)] * (4x^2 + 7x -2) / (4x^2+2x)
= [- 8x / (4x+1)(1-4x)] * (4x - 1)(x + 2) / [2x (2x + 1)]
= - 8x(4x - 1)(x + 2) / [(4x+1)(1-4x)] * 2x (2x + 1)]
= 8x(1 - 4x) (x + 2 ) / [2x(1 - 4x) (4x + 1)(2x + 1)]
= 4(x+2) / [(4x + 1)(2x + 1)]
3a)
Simplify 1/3x+2 - 1/3x+1
= [(3x+1) - (3x+2)] / [(3x+2)(3x+1)]
= - 1 / [(3x+1)(3x+2)]
b)
1 / f(x) = - 1 / [(3x+1)(3x+2)]
1 / f(x) = 1 / [- (3x+1)(3x+2)]
f(x) = - (3x+1)(3x+2)
= - (9x^2 + 3x + 6x + 2)
= - (9x^2 + 9x + 2)
= - 9x^2 - 9x - 2
c)
f(x) = - 9x^2 - 9x - 2
= - (9x^2 + 9x + 2)
= - [(3x)^2 + 2(3x)(9/6) + (9/6)^2 + 2 - (9/6)^2]
= - [(3x + 9/6)^2 + 2 - 81/36]
= - [(3x + 9/6)^2 - 1/4]
= - (3x + 9/6)^2 + 1/4
when (3x + 9/6)^2 = 0 , the maximum value of - (3x + 9/6)^2 = 0 ,
so the maximum value of f(x) = - 0 + 1/4 = 1/4