BASIC FORMULAE & IMPORTANT RESULTS

1. (a + b)² = a² + b² + 2ab

2. (a – b)² = a² + b² – 2ab

3. (a + b)² – (a – b)² = 4ab

4. (a + b)² + (a – b)² = 2(a² + b²)

5. (a² – b²) = (a + b) (a – b)

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

7. (a³ + b³) = (a + b) (a² – ab + b²)

8. (a³ – b³) = (a – b) (a²+ ab + b²)

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

10. If a + b + c = 0, then a³ + b³ + c³ = 3abc

11. ( 1 + 2 + 3 + … + n ) = [n ( n + 1 )] divided by 2.

12. ( 1² + 2² + 3² + … + n²) = [ n (n + 1) (2n + 1)] divided by 6.

13. ( 1³ + 2³ + 3³ + … + n³) = [ n² ( n + 1 )²] divided by 4.

Distributive Law: For any three numbers a, b, c , we have

• a × (b + c) = a × b + a × c

• a × (b – c) = a × b - a × c

Division Algorithm Or Euclidean Algorithm: If we divide a given number by another number then,
DIVIDEND = ( DIVISOR × QUOTIENT ) + REMAINDER.


i. [(x)^n - (a)^n] is divisible by (x-a) for all values of n.

ii.[(x)^n - (a)^n] is divisible by (x+a) for all even values of n.

iii.[(x)^n + (a)^n] is divisible by (x+a) for all odd values of n.

LINEAR EQUATIONS : An equality containing an unknown number is called a linear equation, such as x + 4 = 6 , 3x – 5 = 4, etc.

In a linear equation , we can

• Add same number on both sides,

• Subtract same number on both sides,

• Multiply both sides by a same non – zero number,

• Divide both sides by a same non – zero number.

Transposition :
In a linear equation, we can take any number on the other side with its sign changed from + (positive) to – (negative) and from – (negative) to + (positive). This process is called Transposition.