Simplify each of the following expressions a 16x2x4x2 1 b x3

Simplify each of the following expressions: (a) 1-6/x^2+x/4x^-2 -1 (b) x^3-1/3x^2-2x + 3x^2-x-2/9x^2-4

Solution

4.(a) We have [ 1- 6/(x²+x)]/(4x^-2-1) = [(x²+x -6)/( x²+x)]/ (4/x² -1) = [(x²+x -6)/( x²+x)]/[ (4-x²)/x²] =                  [(x²+x -6)/( x²+x)]*[ x²/(4-x²)] = [(x²+x -6)/[x(x+1)]* [x²/(4-x²)] = [(x²+x -6)/(x+1)]* [x/(4-x²)] = [(x+3)(x-2)/(x+1)]* [x/(2-x)(2+x)] = -x(x+3)/(x+2) [ as x-2 = -(2-x)]

(b) We have [(x³-1)/(3x²-2x)]/[ (3x²-x-2)/(9x²-4)] = [(x³-1)/ x(3x-2)]*[ (9x²-4)/ (3x²-x-2)] =                                    [(x-1)(x²+x+1)/x(3x-2)]* [(3x+2)(3x-2)/(x-1)(3x+2)]=[(x-1)/(x-1)]* [(x²+x+1)/x]*[(3x+2)(3x-2)/(3x-2)(3x+2)] =1*[(x²+x+1)/x]*1 = (x²+x+1)/x.