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 Depdendent Variable

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 Dependent Variable

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# 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 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 