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Remarkable identities

We call remarkable identities to some binomial products that appear very often in calculations with algebraic expressions.


- Square of an addition: (a + b)2 = a2 + b2 + 2ab


(a + b)2 = (a + b)·(a + b) = a2 + ab + ba + b2 = a2 + b2 + 2ab


Example:


(x + 2)2 = x2 + 22 + 2 · x · 2 = x2 + 4x + 4

- Square of a subtraction: (a - b)2 = a2 + b2 - 2ab


(a - b)2 = (a - b)·(a - b) = a2 - ab - ba + b2 = a2 + b2 - 2ab


Example:

 

(2x - 3)2 = (2x)2 + 32 - 2 · 2x · 3 = 4x2 - 12x + 9


- Addition multiplied by subtraction: (a + b)·(a – b) = a2 - b2
    

(a + b)·(a – b) = a2 - ab + ba + b2 = a2 – b2
    

Example:
    

(x + 7)·(x – 7)= x2 – 72 = x2 - 49

 

Exercises:

1.- Expand these expressions:

a) (2x + 1)2 =

b) (3x - 2)2 =

c) (2x - 7)·(2x + 7) =

 

2.- Decompose these polynomials in factors:

a) x2 - 4x + 4 =

b) 4x2 - 1 =

c) 36a2 - 12ab + b2 =

 

 

Solutions: 1.- a) 4x2 + 4x +1; b) 9x2 - 12x + 4; c) 4x2 - 49; 2.- a) (x - 2)2; b) (2x +1)·(2x -1); c) (6a - b)2