Algebraic Transformations: Outline

Algebraic Transformations

Algebraic Expressions

Basic Definitions

Polynomials

Algebraic Transformations

Outline

Factoring

Factoring Quadratic Polynomials

Factoring Cubic Polynomials

Factor Theorems

Other Transformations

Expanding

Completing Perfect Square

Rationalizing Denominators

Algebraic Transformations

Key Topics Remaining: Factoring » Factoring Quadratic Polynomials » Factoring Cubic Polynomials » Factor Theorens » Expanding » Completing Perfect Square » Rationalizing Denominators

An algebraic expression can be represented by many different forms, e.g., the quadratic polynomial x2 – 4x + 3 can be written in the form of

x2 – 4 x + 3 = (x – 2)2 – 1 (*)

as well as the product of two factors:

x2 – 4 x + 3 = (x – 1) (x – 3) (**)

Form (*) allows, for example, to analyze easier the behaviour of the function

y = x2 – 4x + 3,

while form (**) is the best suitable for solving the equation

x2 – 4x + 3 = 0,

Each of the above forms is the "simplest form", depending on a solving problem.
If to consider more complicated expressions including, for instance, several variables, the variety of possible forms becomes still greater.

The following advices can be recommended to perform simplifications of a given expression:

Try different transformations of the expression until you obtain a few forms which are more suitable for the considered problem.

Selecting the simplest form you found, give priority to the expression involving the smallest number of members.

Simplify the expression by factoring or expanding some parts of the expression.

Combine similar terms, collect together terms involving the same powers or radicals, take out common factors, etc.

To simply a fraction, cancel common factors in the numerator and denominator.

Rationalize the denominator.

Reduce the sum of fractions to a common denominator.

Remember, however, that some mathematical problems are solved by decomposition of a compound fraction into the sum of partial fractions.