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ALGEBRA ALGEBRA = 4. Factorise : x - x Sol. x - x = x(x - 1) = x{(x²)2 - 1] = x(x + 1) (x - 1) = x(x + 1) (x + 1) (x - 1) 5. Factorise : x2 - y2 – 2y-1 Sol. x2 - y2 - 2y-1 heads = x2 - (y2 + 2y + 1) x - y + 1)2 (x + y + 1) (x - y - 1) = = 2xla– 1) - 1 6. Factorise : 4x2 *** a + sas a2_1 Sol. 4x2- - 2x 1 a Example : Factorise : a3 - 8b3 + 6403 + 24abc Sol.: a-8b3 + 6403 + 24abc = a + (-25) + (4c)3 - 3ax (-26) (4c) = x + y + 7 - 3xyz Where, a= x, -2b = y and 4c = 2 = (x + y + 2) (x + y2 + z2 - xy - yz - zx) = (a +(-2b) + 4c] (a+ (-2b)2 + (4c)2- a (-2b) -(-2b) (40) - a(4c)] = (a-2b + 4c) (a + 4b2 + 16c2 + 2ab + 8bc-4ac) Example : Factorise : a-b3 + 1 + 3ab Sol. a - b3 + 1 + 3ab = a + (-b)3 + (1)3 - 3a (-b) x 1 = (a - b + 1) (a + b2 + 1 + ab + b-a) Example : Factorise :(p-q) + (q - 113 + (r- p)3 Sol. : If (p-q) = x, (q-1) = y and r-p= 2, then, x + y +2=p-q+q-r+r-p = 0 :: x + y2 + z = 3xyz = 3 (p-9) (q-1) (r-p) Example : If p=2- a, find the value of, a' + bap + po-8 Sol. p= 2-a=a+p+ (-2) = 0 = x + y + z = 0 Where, a= x, p= y and -2 = 2 x + y + z = 3xyz = a + p2 + (-2)3 = 3 x ax px (-2) = a + bap + p 3 - 8 = 0 OTHER SOLVED EXAMPLES 1. Factorise : x2-1a + 1x +1 2x 4x2 - 2xa + .-1 Sky EU = a 1 = 2x (2x - a) + (2x - a) a = + (2x++) = (2x - a) 7. Factorise : x + ab- (a? - 3ab + 4b) Sol. x2 + ab- a² + 3ab-4b2 = x - a² + 4ab-4b2 = x2 - (a²-4ab + 462) = x - (a-2b) (x + a-2b) (x - a + 2b) 8. Factorise : 4x2 + 12xy + 5y = ...
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