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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 ·2 x · 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)

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

Exercises:

1) Calculate:

a) (x - 2)2 - (x - 2)·(x + 2)

b) (2x + 3)2 - 3x2 - (x - 3)2 =

2) Decompose in factors:

a) x2 - 12x + 36 =

b) 9a2 - 16b4 =

Solutions: 1) a) -4x + 8; b) 18x; 2) a) (x - 6)2; b) (3a - 4b2)·(3a + 4b2)