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Rational Expressions and Their Simplification
Radical Expressions and Equations
Algebraic Expressions
Simplifying Algebraic Expressions
Rational Expressions and Functio
Rational Expressions and Functions
Radical Expressions
Rational Expressions Worksheet
Adding and Subtracting Rational Expressions
Rational Expressions
Multiplying and Dividing Rational Expressions
Dividing Fractions, Mixed Numbers, and Rational Expressions
Multiplying and Dividing Rational Expressions
Multiplying and Dividing Rational Expressions
Simplifying Rational Expressions
Complex Rational Expressions
Rational Expressions and Equations
Integration of Polynomial Rational Expressions
Algebraic Expressions
Radical Expressions & Radical Functions
Rational Class and Expression Evaluator
Adding and Subtracting Rational Expressions
Rational Expressions
Radical Expressions
Multiplying Rational Expressions
Rational Expressions and Common Denominators
rational expressions
Polynomial Expressions
Rational Functions, and Multiplying and Dividing Rational Expressions
Simplifying Radical Expressions
Adding and Subtracting Rational Expressions
Rational Expressions and Equations
Rational Expressions
RATIONAL EXPRESSIONS II
Simplifying Expressions
Quadratic Expressions,Equations and Functions
RATIONAL EXPRESSIONS
Absolute Value and Radical Expressions,Equations and Functions
Rational Expressions & Functions

Algebraic Expressions

Sets
Definition. A set is a collection of objects, and these objects are called the elements of
the set.
If S is a set, then a ∈S means that a is an element of S, and means that b is not
an element of S.

Describing Sets
(1) Listing all its elements between curly brackets: S = {1, 2, 3, 4, 5}.
(2) If the elements of a set have a certain property, we can describe the set in terms of
a generic variable that has that property.

Example. which is read as A is the set of all x such that x is greater than 3 .

Definitions
A variable is a letter that can represent any number from a given set of numbers.
When variables such as x, y, and z and some real numbers, and combined using addition,
subtraction, multiplication, division, powers, and roots, we obtain an algebraic expression.
The domain of an algebraic expression is the set of all real numbers that might represent
the variables (that is numbers for which denominators are not zero and roots always exist).

Definition A polynomial in the variable x is an expression of the form

where are real numbers, and n is a nonnegative integer. If , then the
polynomial has degree n. Note that the degree of a polynomial is the highest power of the
variable that appears in the polynomial. The monomials that make up the polynomial
are called the terms of the polynomial.

Example

Adding, Subtracting and Multiplying Polynomials
Examples
Perform the indicated operations and simplify

Solution. To obtain the sum of two polynomials in x we add coefficients of like powers of x.

remove parentheses

add coefficients of like powers of x

simplify

When multiplying two polynimials we use the distributive properties.

Product Formulas
If A and B are any real numbers or algebraic expressions, then

Examples Evaluate the expressions

Solution.

using product formula 1

Factoring Formulas
If A and B are any real numbers or algebraic expressions, then

The first step in factoring expressions is to factor out the common factors.
Example.

1.4 Rational Expressions

Definition. A rational expression is the quotient of two polynomials.
Examples Perform the indicated operation and simplify:

1. Products and quotients.

Solution.

factoring out the factors 2 and 3

 

using the factoring formula 1

simplifying the factors x - 1 and x - 2

2. Sums and differences:

Solution. The denominators are already in factored form. The lcd is

3. Rationalizing a denominator:

Solution.

multiply the numerator and the denominator by
the conjugate of