Course Notes: Fundamentals of Algebra (Exponents, Radicals, Polynomials, Factoring, Rational Expressions)

Vocabulary flashcards covering key terms from the lecture notes.

Exponent

A number that indicates how many times a base is multiplied by itself.

Base

The number or variable that is raised to a power.

Product Rule (Exponents)

When multiplying powers with the same base, add exponents: a^m · a^n = a^(m+n).

Quotient Rule (Exponents)

When dividing powers with the same base, subtract exponents: a^m / a^n = a^(m−n).

Power Rule

Raising a power to another power multiplies the exponents: (a^m)^n = a^(mn).

Zero-Exponent Rule

Any nonzero base raised to the 0th power equals 1: a^0 = 1 (a ≠ 0).

Negative-Exponent Rule

A negative exponent represents a reciprocal: a^(−n) = 1 / a^n (a ≠ 0).

Radical

A root expression, such as the n-th root of a: √[n]{a}.

Radicand

The number inside a radical sign.

Index

The root degree; 2 for square root, 3 for cube root, etc.

Rational Exponent

An exponent written as a fraction p/q; a^(p/q) = qth root of a^p.

Simplified Radical Form

A radical expression with no factors under the radical that can be extracted or simplified further.

Perfect Square

A number that is the square of an integer.

Rationalize Denominator

Process of removing a radical from the denominator by multiplying by a suitable factor.

Product Rule for Radicals

For nonnegative radicands, √a · √b = √(ab).

Quotient Rule for Radicals

√(a/b) = √a / √b (b > 0, a ≥ 0 in typical contexts).

Radical Equation

An equation that contains radicals.

Monomial

A polynomial with exactly one term.

Binomial

A polynomial with exactly two terms.

Trinomial

A polynomial with exactly three terms.

Polynomial

An expression formed by a sum of monomials with nonnegative integer exponents.

Standard Form (Polynomial)

Polynomial written in descending powers of the variable.

Degree

The greatest exponent among the terms of a polynomial.

FOIL

First, Outside, Inside, Last—method to multiply two binomials.

GCF (Greatest Common Factor)

The largest factor that divides all terms of a polynomial.

Factor by Grouping

A factoring method that groups terms to factor common factors.

Difference of Squares

a^2 − b^2 = (a − b)(a + b).

Perfect Square Trinomial

A trinomial in the form a^2 ± 2ab + b^2 = (a ± b)^2.

Sum/Difference of Cubes

a^3 ± b^3 = (a ± b)(a^2 ∓ ab + b^2).

Factor a Trinomial

Factor ax^2 + bx + c into a product of binomials.

Factor Completely

Express a polynomial as a product of irreducible factors.

Rational Expression

A fraction whose numerator and/or denominator is a polynomial.

Excluded Value (Domain)

A value that makes a denominator zero or is otherwise not allowed in a rational expression.

Complex Fraction

A fraction in which the numerator or denominator itself contains a fraction.

Scientific Notation

Writing numbers as a × 10^n with 1 ≤ |a| < 10.

Decimal Notation

The standard base-10 representation of numbers with a decimal point.

Prime Factorization

Expressing a number as a product of prime numbers.