Questions and Answers
What are natural numbers?
What are whole numbers?
What are integers?
What is a rational number?
Signup and view all the answers
What is an irrational number?
Signup and view all the answers
What do real numbers consist of?
Signup and view all the answers
What are closure properties in mathematics?
Signup and view all the answers
What are commutative properties?
Signup and view all the answers
What do associative properties refer to?
Signup and view all the answers
What is the identity property?
Signup and view all the answers
What are inverse properties?
Signup and view all the answers
What does the distributive property state?
Signup and view all the answers
What is the order of operations?
Signup and view all the answers
What is the absolute value?
Signup and view all the answers
Define absolute value by the rules.
Signup and view all the answers
What are the properties of absolute value?
Signup and view all the answers
What are the rules for positive exponents?
Signup and view all the answers
What is a binomial?
Signup and view all the answers
What is a trinomial?
Signup and view all the answers
How do you determine if an expression is a polynomial?
Signup and view all the answers
What are special products in polynomials?
Signup and view all the answers
What is GCF?
Signup and view all the answers
What is factoring?
Signup and view all the answers
When is a polynomial prime?
Signup and view all the answers
When is a polynomial factored completely?
Signup and view all the answers
How could you factor a second-degree trinomial?
Signup and view all the answers
What are the special products?
Signup and view all the answers
Study Notes
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).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
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.
