1. Cubic Identities

2. Summary
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a - b)3 = a3 - 3a2b + 3ab2 - b3
(x + a) (x + b) (x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc
Deduced Identities
a3 + b3 = (a + b) (a2 – ab + b2)
a3 - b3 = (a - b) (a2 + ab + b2)

3. Example 1:-
Find the value of (3m + 5) (3m + 5) (3m + 5) using suitable identity
Solution:
The above algebraic expression is same as following identity
(a + b)3 = (a + b) (a + b) (a + b) = a3 + 3a2b + 3ab2 + b3
(3m + 5) (3m + 5) (3m + 5) Where a = 3m and b = 5
a3 + 3a2b + 3ab2 + b3
=(3m)3 + 3 x (3m)2 x x 3m x (After substituting value of a and b)
= 27m3 + 3 x 9m2 x x 3m x
= 27m m m (Ans)

4. Example 2:- Solution: a3 - 3a2b + 3ab2 - b3
Find the value of (2x - 3) (2x - 3) (2x - 3) using suitable identity
Solution:
The above algebraic expression is same as following identity
(a - b)3 = (a - b)(a - b)(a - b) =a3 - 3a2b + 3ab2 - b (2x - 3) (2x - 3) (2x - 3) Where a = 2x and b = 3
a3 - 3a2b + 3ab2 - b3
=(2x)3 -2 x (2x)2 x x 2x x (After substituting value of a and b)
= 8x3 - 2 x 4x2 x x 2x + 27
= 8x3 – 24x2 + 18x (Ans)

5. The above algebraic expression is same as following identity
Example 3:-
Find the value of (5m + 2) (5m + 4) (5m + 6) using suitable identity
Solution:
The above algebraic expression is same as following identity
(x + a) ( x + b) ( x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc
(5m + 2) (5m + 4) (5m + 6) Where x= 5m , a=2 , b= 4 and c = 6
x3 + (a + b + c)x2 + (ab + bc + ca)x + abc
= (5m)3 + ( ) x (5m)2 + (2x4 + 4x6 + 6x2) x (5m) + 2 x 4 x (After substituting value of x, a ,b and c)
= 125m x 25m2 + ( ) x 5m + 48
= 125m m m (Ans)

6. Try these Using identity solve (2m + 3) (2m + 3) (2m + 3)
(3x – 5) (3x – 5) (3x – 5)
(8x + 5) (8x + 2) (8x + 4)