Fundamental Concepts of Algebra: Precalculus Study Notes

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Fundamental Concepts of Algebra

Algebraic Expressions

Algebraic expressions are foundational in algebra and precalculus, representing combinations of numbers, variables, and mathematical operations. Understanding their structure is essential for manipulating and evaluating expressions.

Algebraic Expression: A combination of numbers and variables connected by mathematical operations (addition, subtraction, multiplication, division, exponents).
Variable: A letter representing an unknown or changeable value, typically denoted by x, y, or z.
Coefficient: The numerical factor multiplying a variable (e.g., in 3x, 3 is the coefficient).
Constant: A number without variables; its value does not change (e.g., in 3x + 5, 5 is the constant).

Example: In the expression 4x + 7, 4 is the coefficient, x is the variable, and 7 is the constant.

When an equals sign (=) is present, the expression becomes an equation.

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute the given values for the variables and follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

Example: Evaluate 2x + 5 when x = 3:

Order of Operations

Parentheses
Exponents
Multiplication/Division
Addition/Subtraction

Always apply these rules when simplifying or evaluating expressions.

Simplifying Algebraic Expressions

Simplifying involves reducing expressions to their simplest form by combining like terms and applying distributive properties.

Term: Parts of an expression separated by + or - signs.
Like Terms: Terms with the same variable(s) raised to the same power(s).

Steps to Simplify:

Distribute constants/variables through parentheses.
Group like terms together.
Combine like terms by addition or subtraction.

Example: Simplify :

Rules of Exponents

Exponent Properties

Exponent rules are essential for simplifying expressions involving powers.

Name	Rule	Description
Product Rule		Add exponents when multiplying like bases.
Quotient Rule		Subtract exponents when dividing like bases.
Power Rule		Multiply exponents when raising a power to a power.
Zero Exponent		Any nonzero base raised to zero is 1.
Negative Exponent		Negative exponent means reciprocal.
Power of a Product		Distribute exponent to each factor.
Power of a Quotient		Distribute exponent to numerator and denominator.

Example: Simplify :

Introduction to Polynomials

Definition and Classification

A polynomial is an algebraic expression where variables have only whole number exponents (no negatives or fractions).

Monomial: One term (e.g., )
Binomial: Two terms (e.g., )
Trinomial: Three terms (e.g., )

Standard Form: Terms are written in descending order of exponents, and like terms are combined.

Degree: The highest exponent of the variable in the polynomial.
Leading Coefficient: The coefficient of the term with the highest degree.

Example: is a trinomial of degree 3 with leading coefficient 6.

Adding and Subtracting Polynomials

Combine like terms to add or subtract polynomials.

Example:

Multiplying Polynomials

Use the distributive property or the FOIL method (First, Outer, Inner, Last) for binomials.

Example (FOIL):

For polynomials with more than two terms, distribute each term in one polynomial to every term in the other.

Special Products

Square of a Binomial:
Difference of Squares:
Cube of a Binomial:

Factoring Polynomials

Factoring Methods

Greatest Common Factor (GCF): Factor out the largest common factor from all terms.
Grouping: Group terms into pairs and factor each group, then factor out the common binomial.
Special Product Formulas: Use formulas for perfect square trinomials, difference of squares, and sum/difference of cubes.
AC Method: For quadratics , find two numbers that multiply to and add to .

Example (GCF):

Example (Difference of Squares):

Radical Expressions

Square Roots and nth Roots

The square root of a number is the value that, when multiplied by itself, gives the original number. The nth root generalizes this concept.

Principal Root: The positive root.
Negative Root: The negative root.
Radicand: The term inside the radical symbol.

Example: , , is imaginary.

Simplifying Radical Expressions

Rewrite the radicand as a product where one factor is a perfect power.
Apply the property .

Example:

Radicals with Variables

Apply the same rules to variables: (for ).

Example:

Radicals with Fractions

Use to split or combine radicals.

Example:

Adding and Subtracting Radicals

Combine only like radicals (same radicand and index).

Example:

Rationalizing Denominators

Radicals should not remain in the denominator. Multiply numerator and denominator by a value that eliminates the radical.

Example:

For denominators with two terms, multiply by the conjugate.

Example:

Rational Exponents

Radicals can be rewritten as exponents with fractional powers.

Example: