College Algebra Module 1 Flashcards

Number Sets

• Natural Numbers: Counting numbers including {1, 2, 3, 4,…}.
• Whole Numbers: Natural numbers plus zero, represented as {0, 1, 2, 3, 4,…}.
• Integers: Positive and negative whole numbers including zero, valued as {-3, -2, -1, 0, 1, 2, 3,…}.
• Rational Numbers: Any integer, repeating, or terminating decimals; includes fractions.
• Irrational Numbers: Real numbers that cannot be expressed as a fraction; examples include π and square roots not resulting in whole numbers.
• Real Numbers: Combination of both rational and irrational numbers.

Mathematical Properties

• Closure Properties: For any real numbers a and b, both a + b and a * b result in a real number.
• Commutative Properties: Addition (a + b = b + a) and multiplication (a * b = b * a) are order-independent.
• Associative Properties: Grouping of numbers doesn't affect the result for addition (a + (b + c) = (a + b) + c) or multiplication (a * (b * c) = (a * b) * c).
• Identity Properties: Adding zero (a + 0 = a) or multiplying by one (a * 1 = a) keeps the original number unchanged.
• Inverse Properties: Adding the negative (a + (-a) = 0) or multiplying by the reciprocal (a * (1/a) = 1) yields an identity.
• Distributive Property: a(b + c) = ab + ac indicates how to distribute multiplication over addition.

Order of Operations

• Operate fractions above and below the bar separately.
• Work within parentheses from innermost to outermost.
• Simplify powers and roots from left to right.
• Perform multiplication and division consecutively from left to right.
• Complete addition and subtraction last, in sequence from left to right.
• Acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

Absolute Value

• Represents the distance from zero on a number line.
• Defined by rules: IaI = a if a ≥ 0, and IaI = -a if a < 0.
• Properties of Absolute Value:
  • Always non-negative (IaI ≥ 0).
  • The absolute value of negatives equals the absolute value of positives (I-aI = IaI).
  • Product and quotient of absolute values conform to certain rules: IaI * IbI = IabI and IaI / IbI = Ia/bI.
  • The triangle inequality states Ia + bI ≤ IaI + IbI.

Exponent Rules

• For positive exponents: apply product rule (a^m * a^n = a^(m+n)), power rule ((a^m)^n = a^(mn)), and others including (ab)^m = a^m * b^m and (a/b)^m = a^m / b^m.
• Zero Exponent Rule: Any non-zero base raised to the zero power equals one (a^0 = 1).

Polynomials

• Binomial: A polynomial consisting of exactly two terms.
• Trinomial: A polynomial with exactly three terms.
• Polynomial Determination: To qualify, expressions must consist of real numbers and all degrees should be whole numbers.
• Special Products:
  • Difference of squares: x^2 - y^2 = (x+y)(x-y)
  • Perfect squares: x^2 + 2xy + y^2 = (x+y)^2 and x^2 - 2xy + y^2 = (x-y)^2.

Factoring Polynomials

• Greatest Common Factor (GCF): The largest common product of numbers, variables, or expressions across all polynomial terms utilized in factoring.
• Factoring Process: Identifying polynomials whose product equals a given polynomial.
• Prime Polynomial: A polynomial that cannot be expressed as a product of two other polynomials (excluding -1 and 1); sums of squares are particularly noted as prime.
• Complete Factorization: A polynomial is fully factored when expressed only as a product of prime polynomials.
• Factoring a Second-Degree Trinomial: Reverse FOIL method used to express ax^2 + bx + c into two binomials.

Special Products Reminder

• Difference of squares: x^2 - y^2 = (x - y)(x + y).
• Perfect square trinomial: x^2 + 2xy + y^2 = (x + y)^2.
• Difference of cubes: x^3 - y^3 = (x - y)(x^2 + xy + y^2).
• Sum of cubes: x^3 + y^3 = (x + y)(x^2 - xy + y^2).

Test your understanding of fundamental concepts in College Algebra with these flashcards. This module covers key definitions including natural numbers, whole numbers, integers, and types of numbers like rational and irrational. Perfect for students looking to reinforce their knowledge in beginning algebra.