IDENTITIES
Identity : It is an equally which is true for all values of variables.
SOME FORMULAE
- product of an identity (x + a) (x + b) = x2 + (a + b) x + ab Now when b = a then (x + a) (x + a) = x2 + 2ax + a2
when b = – a then (x + a) (x – a) = x2 – a2 when b = – b we have x2 + (a – b) x – ab
when a = – a,b = – b we have = x2 – (a + b) x + ab
- product of (x + y) (x2 – xy + y2) = x3 + y3
- product of (x – y) (x2 + xy + y2) = x3 – y3
- product of (x + y + z) (x2 + y2 + z2 – xy – yz – xz) = x3 + y3 + z3 – 3xyz
- Cube of a binomial
(x + y)3 = x3 + y3 + 3xy(x + y) x3 + y3 = (x + y)3 – 3xy(x + y) (x – y)3 = x3 – y3 – 3xy (x – y)
x3 – y3 = (x – y)3 + 3xy (x – y)
- Square of a trinomial
(x + y + z)2 = y2 + y2 + z2 + 2xy + 2yz + 2xz (1) (x – y + z)2 = x2 + y2 + z2 + 2xy – 2xz – 2yz (2) (-x -y + z)2 = x2 + y2 + z2 – 2xy -2yz + 2xz (3) (-x + y + z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz (4) (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
(5) (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz If x + y + z = 0 then x3 + y3 + z3 = 3xyz.
(6) x2 – y2 = (x + y) (x – y)
(7) x4 – y4 = (x – y) (x2 + x2y + xy2 + y3)
(8) x5 + y5 = (x + y) (x4 – x3y + x2y2 – xy3 + y4)
