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