1.               (a + b)(a – b) = a2 – b2

2.               (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

3.               (a ± b)2 = a2 + b2± 2ab

4.               (a + b + c + d)2 = a2 + b2 + c2 + d2 + 2(ab + ac + ad + bc + bd + cd)

5.               (a ± b)3 = a3 ± b3 ± 3ab(a ± b)

6.               (a ± b)(a2 + b2 m ab) = a3 ± b3

7.               (a + b + c)(a2 + b2 + c2 -ab – bc – ca) = a3 + b3 + c3 – 3abc =

1/2 (a + b + c)[(a - b)2 + (b - c)2 + (c - a)2]

8.               when a + b + c = 0, a3 + b3 + c3 = 3abc

9.               (x + a)(x + b) (x + c) = x3 + (a + b + c) x2 + (ab + bc + ac)x + abc

10.           (x – a)(x – b) (x – c) = x3 – (a + b + c) x2 + (ab + bc + ac)x – abc

11.           a4 + a2b2 + b4 = (a2 + ab + b2)( a2 – ab + b2)

12.           a4 + b4 = (a2 – √2ab + b2)( a2 + √2ab + b2)

13.           an + bn = (a + b) (a n-1 – a n-2 b +  a n-3 b2 – a n-4 b3 +…….. + b n-1)
(valid only if n is odd)
14.           an – bn = (a – b) (a n-1 + a n-2 b +  a n-3 b2 + a n-4 b3 +……… + b n-1)
{where n ϵ N)
15.           (a ± b)2n is always positive while -(a ± b)2n is always negative, for any real values of a and b

16.           (a – b)2n = (b – a)2” and (a – b)2n+1 = – (b – a)2n+1

17.           if α and β are the roots of equation ax2 + bx + c = 0, roots of cx” + bx + a = 0 are 1/α and 1/β.
if α and β are the roots of equation ax2 + bx + c = 0, roots of ax2 – bx + c = 0 are -α and -β.

18.
o                n(n + l)(2n + 1) is always divisible by 6.

o                32n leaves remainder = 1 when divided by 8

o                n3 + (n + 1 )3 + (n + 2 )3 is always divisible by 9

o                102n + 1 + 1 is always divisible by 11

o                n(n2- 1) is always divisible by 6

o                n2+ n is always even

o                23n-1 is always divisible by 7

o                152n-1 +l is always divisible by 16

o                n3 + 2n is always divisible by 3

o                34n – 4 3n is always divisible by 17

o                n! + 1 is not divisible by any number between 2 and n
(where n! = n (n – l)(n – 2)(n – 3)…….3.2.1)
for eg 5! = 5.4.3.2.1 = 120 and similarly 10! = 10.9.8…….2.1= 3628800

19.           Product of n consecutive numbers is always divisible by n!.

20.           If n is a positive integer and p is a prime, then np – n is divisible by p.

21.           |x| = x if x ≥ 0 and |x| = – x if x ≤ 0.

22.           Minimum value of a2.sec2Ɵ + b2.cosec2Ɵ is (a + b)2; (0° < Ɵ < 90°)

for eg. minimum value of 49 sec2Ɵ + 64.cosec2Ɵ is (7 + 8)2 = 225.
23.           among all shapes with the same perimeter a circle has the largest area.

24.           if one diagonal of a quadrilateral bisects the other, then it also bisects the quadrilateral.

25.           sum of all the angles of a convex quadrilateral = (n – 2)180°

26.           number of diagonals in a convex quadrilateral = 0.5n(n – 3)

27.           let P, Q are the midpoints of the nonparallel sides BC and AD of a trapezium ABCD.Then,
ΔAPD = ΔCQB.