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).