user
Simplify 1/x^2 - x^2
alphonsio

The expression 1x2−x2\frac{1}{x^2} - x^2 can not be simplified, therefore, it can be expressed as :

1−x4x2\frac{1 - x^4}{x^2}

or

(1−x2)(1+x2)x2\frac{(1 - x^2)(1 + x^2)}{x^2}

The expression 1x2−x2\frac{1}{x^2} - x^2 combines two different terms that involve the variable xx.

  • The first term, 1x2\frac{1}{x^2}, is the reciprocal of the square of xx.
  • The second term, x2x^2, is the square of xx.

These terms cannot be simplified together algebraically since one is rational (a fraction) and the other is polynomial. Therefore, the expression remains as is:

1x2−x2\frac{1}{x^2} - x^2

When expressing everything with a common denominator:

1x2−x4x2=1−x4x2\frac{1}{x^2} - \frac{x^4}{x^2} = \frac{1 - x^4}{x^2}

Since 1−x41 - x^4 can be factored as:

1−x4=(1−x2)(1+x2)1 - x^4 = (1 - x^2)(1 + x^2)

The expression then simplifies to:

(1−x2)(1+x2)x2\frac{(1 - x^2)(1 + x^2)}{x^2}

So the final simplified form is:

(1−x2)(1+x2)x2\frac{(1 - x^2)(1 + x^2)}{x^2}

In conclusion :

1x2−x2=1−x4x2=(1−x2)(1+x2)x2\frac{1}{x^2} - x^2 = \frac{1 - x^4}{x^2} =\frac{(1 - x^2)(1 + x^2)}{x^2}