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 and manipulation is essential for solving equations and modeling real-world problems.

• 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 5x, 5 is the coefficient).

• Constant: A number without variables; its value does not change (e.g., in 5x + 3, 3 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

This order ensures consistent and correct evaluation of expressions.

Exponents in Expressions

Exponents represent repeated multiplication of a base. They are written as a small number (the exponent) to the upper right of the base.

• Base: The number being multiplied.

• Exponent (Power): Indicates how many times the base is multiplied by itself.

Example:

Evaluate 43:

Simplifying Algebraic Expressions

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

• 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:

  1. Distribute constants/variables through parentheses.

  2. Group like terms together.

  3. Combine like terms by addition or subtraction.

Example: Simplify :

Rules of Exponents

Exponent rules allow for the simplification and manipulation of expressions involving powers.

Zero & Negative Exponents

• Zero exponent: (for )

• Negative exponent:

Simplifying Expressions with Exponents

• Apply exponent rules in logical order, usually from the innermost parentheses outward.

• Fully simplified expressions have no powers raised to powers, no parentheses, no like bases multiplied or divided, no zero or negative exponents, and all operations performed.

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 of Polynomials

• Write terms in descending order of exponents.

• Combine like terms.

• Degree: The highest exponent of the variable.

• Leading Coefficient: The coefficient of the term with the highest degree.

Example: is in standard form; degree is 3, leading coefficient is 6.

Adding & Subtracting Polynomials

• Add or subtract like terms (same variable and exponent).

• Distribute negative signs when subtracting.

Example:

Multiplying Polynomials

• FOIL Method: Used for multiplying two binomials: First, Outer, Inner, Last.

• Distributive Property: Multiply each term in one polynomial by each term in the other.

Example (FOIL):

Example (Distributive):

Special Products

• Square of a Binomial:

• Difference of Squares:

• Cube of a Binomial:

Factoring Polynomials

Factoring Out the Greatest Common Factor (GCF)

• Identify the largest factor common to all terms and factor it out.

Example:

Factoring by Grouping

• Group terms into pairs, factor out the GCF from each group, then factor out the common binomial.

Example:

Factoring Using Special Product Formulas

• Recognize and apply formulas for perfect square trinomials, difference of squares, and sum/difference of cubes.

Example:

Factoring Using the AC Method

• For quadratics , find two numbers that multiply to and add to .

• Rewrite the middle term, group, and factor.

Example:

Radical Expressions

Square Roots and nth Roots

• The square root of a number is a value such that .

• Positive real numbers have two square roots: principal (positive) and negative.

• For even roots, negative radicands yield imaginary numbers; for odd roots, negatives are allowed.

Example: , , is imaginary.

Simplifying Radical Expressions

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

• Apply the property .

Example:

Simplifying Radicals with Variables and Fractions

• Apply radical rules separately to numbers and variables.

• For fractions:

Example:

Adding and Subtracting Radicals

• Combine only like radicals (same radicand and index).

• Simplify unlike radicals before combining if possible.

Example:

Rationalizing Denominators

Radicals should not remain in the denominator of a fraction. To rationalize, multiply numerator and denominator by a value that eliminates the radical.

• For single-term denominators, multiply by the radical itself.

• For two-term denominators, multiply by the conjugate (change the sign between terms).

Example:

Example (Conjugate):

Rational Exponents

Radical expressions can be rewritten using rational (fractional) exponents.

• Exponent rules apply to rational exponents as well.

Example: