# 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

We can use the remarkable identities:

- In calculations:

(x +1)2 – (x – 1)2 = x2 + 2x + 1 – (x2 – 2x + 1)= x2 + 2x + 1 – x2 + 2x - 1= 4x

- To decompose a polynomial in factors :

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

x2 -  9 = (x + 3)·( x – 3)

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 =

3.- Calculate:

a) (x + 2)2 - (x + 2)·(x -2) =

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

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

3.- a) 4x + 8; b) 4ab -b2