Question

Упростить (X+1/X)^2×(X^-2+2x^-1+1)

Answer 1

=(х²+2+1/х²)(1/х²+2/х+1)=1+2х+х²+2/х²+4/х+2+1/х^4 +2/х³+1/х²=
х²+2х+1/х^4+2/х³+3/х²+4/х+3

Answer 2

Let's simplify the expression step by step:

1. Expand the first term in parentheses using the formula (a + b)^2 = a^2 + 2ab + b^2:

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(X + 1/X)^2 = X^2 + 2(X)(1/X) + (1/X)^2
            = X^2 + 2 + 1/X^2

2. Expand the second term in parentheses using a common denominator of X^-2:

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X^-2 + 2X^-1 + 1 = X^-2 + 2(X/X^2) + (X^2/X^2)
                  = X^-2 + 2X^-1 + X^0/X^2
                  = X^-2 + 2X^-1 + 1/X^2

3. Substitute the expanded terms back into the original expression and simplify:

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(X+1/X)^2 × (X^-2+2x^-1+1) 
= (X^2 + 2 + 1/X^2) × (X^-2 + 2X^-1 + 1/X^2)
= X^2(X^-2) + 2X^2(X^-1) + X^2(1/X^2) 
  + 2(X)(1/X^-1) + 2(X)(2X^-2) + 2(1/X^2)
  + (1/X^2)(X^-2) + 2(X^-2) + 1/X^2(1/X^2)
= 1 + 2X + 2/X + 2X^-1 + 3X^-2 + 1/X^2
= 3 + (X + 1/X)^2 + 2(X + 1/X)
= 3 + X^2 + 2 + 2X + 1/X^2 + 2X + 2/X
= X^2 + 4X + 3 + 2/X^2 + 2/X

Therefore, the simplified expression is: X^2 + 4X + 3 + 2/X^2 + 2/X.