Question

If x = √2 – 1, find the value of x4 + 4x3 + 6x2 + 4x + 7.

Answer 1

Answer:

Lets factorize (x^4-4x^3+4x^2 )

Taking x^2 common, we get,

(x^2) (x^2–4x+4)

= (x^2) (x-2)(x-2)

= (x^2) ((x-2)^2 )

Substituting x= 1+√2 in the above expression, we get

= ((1+√2)^2) ((1+√2–2)^2)

=((√2+1)^2)((√2–1)^2)

The above expression can be written as (taking squares common)

=((√2+1)(√2–1))^2

=((√2)^2) -(1^2))^2, since (a+b)(a-b)=a^2 - b ^2

= (2–1)^2 = 1^2

= 1