x4 - 2x3 + 3x2 - 2x + 2
= x4 + 4x2 + 4 - 2x3 - 2x - 2 - x2
= ( x2 + 2)2 - 2x( x2 + 1) - ( x2 + 2)
Đặt : x2 + 2 = a , ta có :
a2 - 2x( a - 1) - a
= a2 - 2ax + 2x - a
= - a( 2x - a) + ( 2x - a)
= ( 2x - a)( 1 - a)
Thay : x2 + 2 = a , ta có :
( 2x - x2 - 2)( 1 - x2 - 2)
= ( 2x - x2 - 2)( - 1 - x2)
x4 + 2x3 + 3x2 + 2x + 1
= x4 + 2x3 + x2 + x2 + x2 + 2x + 1
= x2(x2 + 2x + 1) + (x2 + 2x + 1) + x2
= (x2 + 1)(x + 1)2 + x2 + 1 - 1
= (x2 + 1)(x + 1)2 + (x2 + 1) – 1
= (x2 + 1)[(x + 1)2 + 1] – 1
= \(\left(\sqrt{\left(x^2+1\right)\left[\left(x+1\right)^2+1\right]}\right)^2-1\)
= {(x2 + 1)[(x + 1)2 + 1] - 1}{(x2 + 1)[(x + 1)2 + 1] + 1}
