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