Podcast
Questions and Answers
What are natural numbers?
What are natural numbers?
What are whole numbers?
What are whole numbers?
What are integers?
What are integers?
What is a rational number?
What is a rational number?
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What is an irrational number?
What is an irrational number?
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What do real numbers consist of?
What do real numbers consist of?
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What are closure properties in mathematics?
What are closure properties in mathematics?
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What are commutative properties?
What are commutative properties?
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What do associative properties refer to?
What do associative properties refer to?
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What is the identity property?
What is the identity property?
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What are inverse properties?
What are inverse properties?
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What does the distributive property state?
What does the distributive property state?
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What is the order of operations?
What is the order of operations?
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What is the absolute value?
What is the absolute value?
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Define absolute value by the rules.
Define absolute value by the rules.
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What are the properties of absolute value?
What are the properties of absolute value?
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What are the rules for positive exponents?
What are the rules for positive exponents?
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What is a binomial?
What is a binomial?
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What is a trinomial?
What is a trinomial?
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How do you determine if an expression is a polynomial?
How do you determine if an expression is a polynomial?
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What are special products in polynomials?
What are special products in polynomials?
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What is GCF?
What is GCF?
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What is factoring?
What is factoring?
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When is a polynomial prime?
When is a polynomial prime?
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When is a polynomial factored completely?
When is a polynomial factored completely?
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How could you factor a second-degree trinomial?
How could you factor a second-degree trinomial?
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What are the special products?
What are the special products?
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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.
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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.
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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
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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.
