Operations on Polynomials, Rational Expressions, and Simplification of Expressions

A good way to gain familiarity with Mathematica's operations to do symbolic algebra is to look though the categories in the Built-in Functions section in the Help Browser.  Look though the names of functions  under Algebraic Computation--they have names that are fairly indicative of their purpose.

The coefficient of

Rewrite as a polynomial in x, but simplify the coefficients individually

Rational Expressions

Simplfiying Expressions with Square Roots

One common hurdle is getting Mathematica  to remove roots:

Note in the following statement that "x ∈ Reals" means "x is a real number" and "&&" is the "Logical And" operator.

Brute force; clever , but not a good idea follows:

Created by Wolfram Mathematica 6.0  (28 August 2007)