College Algebra - Real Number System
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Questions and Answers

What is a set?

  • A group or collection of objects (correct)
  • A type of function
  • An equation
  • A mathematical operation
  • What is the empty set also known as?

    null set

    Which is a subset of the natural numbers?

  • Rational numbers
  • Whole numbers (correct)
  • Integers
  • Real numbers
  • Match the following number classifications with their descriptions:

    <p>Natural Numbers = Counting numbers like 1, 2, 3… Whole Numbers = Natural numbers and zero Integers = Whole numbers and negative natural numbers Rational Numbers = Ratios of integers where the denominator is not zero</p> Signup and view all the answers

    Irrational numbers have decimal representations that terminate.

    <p>False</p> Signup and view all the answers

    What are complex numbers?

    <p>Combination of a real number and an imaginary number</p> Signup and view all the answers

    What is the correct order of operations?

    <p>Start with innermost parentheses, then perform all indicated exponents, followed by multiplication and division from left to right, and finally addition and subtraction from left to right.</p> Signup and view all the answers

    What is the simplified expression of 2x + 3(4 - x)?

    <p>12 - x</p> Signup and view all the answers

    What is the simplified expression of -{-3 - 5 - 4x + z - 2y + 2z}?

    <p>60x - 6y - 27z</p> Signup and view all the answers

    What is the simplified expression of -12^2 + m^3 - m - m - 2m - 6?

    <p>12m^2 - 48m + 48</p> Signup and view all the answers

    What defines a polynomial?

    <p>A polynomial is an expression constructed from variables and constants using addition, subtraction, and multiplication, with exponents as non-negative integers.</p> Signup and view all the answers

    What are like terms?

    <p>Terms that contain the same variables with the same powers, such as 3x and 6x.</p> Signup and view all the answers

    What is an example of a polynomial that has a degree of 3?

    <p>4x^3 + 3x^2 + 2x + 1</p> Signup and view all the answers

    What factors disqualify expressions from being polynomials?

    <p>Negative exponents, division by variables, and fractional exponents disqualify expressions from being polynomials.</p> Signup and view all the answers

    What is the degree of the polynomial 2x^3y^2 + 5xy^4 + 3x^2 + 7?

    <p>5</p> Signup and view all the answers

    How do you add polynomials?

    <p>Addition of polynomials is performed by applying the rules of addition on like terms.</p> Signup and view all the answers

    What is the result of adding the polynomials 3x^2 + 2xy - y^2, x^2 - 2xy + 6y^2, and 9x^2 + 11xy - 5y^2?

    <p>13x^2 + 11xy</p> Signup and view all the answers

    What is the outcome of subtracting the first polynomial from the second: 6x + 3xy - 7y and 3x - 2xy + 6y?

    <p>-3x - 5xy + 13y</p> Signup and view all the answers

    What should you do to multiply polynomials?

    <p>Multiply each term of one polynomial by each term of the other to obtain partial products, then combine like terms.</p> Signup and view all the answers

    What is the FOIL method used for?

    <p>The FOIL method is used for multiplying binomials.</p> Signup and view all the answers

    Study Notes

    Sets

    • A set is a collection of distinct objects called members or elements.
    • Example: The starting lineup of a baseball team is a subset of the entire team.
    • If all members of set B are also in set A, then B is a subset of A, denoted as B ⊂ A.
    • The empty set, lacking elements, is denoted by a specific symbol.

    The Real Number System

    • Real numbers include two primary subsets: Rational and Irrational numbers.

    Rational Numbers

    • Comprised of:
      • Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}
      • Whole Numbers: {0, 1, 2, 3, 4, ...}
      • Positive Integers/Natural Numbers: {1, 2, 3, 4, ...}
      • Fractions: Represented as ratios of integers (e.g., 2/3, 5/4).
    • Decimal representation either terminates or repeats.

    Irrational Numbers

    • Numbers with non-terminating, non-repeating decimal representations.
    • Examples include √2, π, and e.

    Symbols and Descriptions

    • N: Natural Numbers (Counting Numbers). Example: {1, 2, 3, 4, 5, ...}
    • W: Whole Numbers (Natural Numbers plus Zero). Example: {0, 1, 2, 3, 4, 5, ...}
    • Z: Integers (Whole Numbers and Negative Whole Numbers). Example: {..., -3, -2, -1, 0, 1, 2, 3, ...}
    • Q: Rational Numbers (Ratios of integers where the denominator is not zero). Example: {..., -19/1, -17, 0, 1.37, 3.666...}
    • I: Irrational Numbers. Examples: √2, π, 1.2179...
    • R: Real Numbers (Combination of Rational and Irrational Numbers). Example: {2, π, -17.25, 7/3}

    Complex Numbers

    • Composed of a real number and an imaginary number.
    • Can be expressed in:
      • Rectangular Form: x + yi or x + ji.
      • Polar Form: r∠θ.
      • Trigonometric Form: r cos θ + j sin θ.
      • Exponential Form: re^(jθ).

    Order of Operations

    • Follow the sequence: Parentheses → Exponents → Multiplication/Division → Addition/Subtraction.
    • Work from the innermost parentheses outward, handling exponents from left to right, and applying multiplication/division then addition/subtraction from left to right.

    Polynomial Terminology

    • A polynomial consists of one or more monomials combined by addition or subtraction.
    • Monomials are products of a constant and a variable raised to a nonnegative integer power (e.g., ( ax^k )).
    • The constant ( a ) is the coefficient, and ( k ) is the degree of the monomial.
    • Like terms are terms with the same variables raised to the same powers, such as ( 3x ) and ( 6x ).

    Characteristics of Polynomials

    • Polynomials contain only variables raised to non-negative integer powers and involve addition, subtraction, and multiplication.
    • Example polynomial expressions include ( 3x^2 + 2x + 1 ) and ( 3x^4 - 2x^3 + 5x^2 - x + 7 ).

    Non-Polynomial Expressions

    • Expressions that do not qualify as polynomials include:
      • Division by a variable (e.g., ( \frac{3}{x} ))
      • Negative or fractional exponents (e.g., ( x^{-1} + 4 ))
      • Trigonometric functions (e.g., ( \sin x + x ))
      • Exponential functions (e.g., ( 2^x + x ))
      • Logarithmic functions (e.g., ( \log x + x ))

    Degree of a Polynomial

    • The degree is the highest power of the variable within the polynomial.
    • For polynomials with multiple variables, the degree is the highest sum of the exponents in any term (e.g., ( 2x^3y^2 + 5xy^4 ) has a degree of 5).

    Addition and Subtraction of Polynomials

    • Combine like terms using addition and subtraction principles.
    • Example:
      • ( 5x^2 - 2x + 3 + (3x^3 - 4x^2 + 7) ) simplifies to ( 3x^3 + x^2 - 2x + 10 ).

    Sample Problems

    • Add polynomials:
      • Example: ( 3x^2 + 2xy - y^2 ), ( x^2 - 2xy + 6y^2 ), and ( 9x^2 + 11xy - 5y^2 ) result in ( 13x^2 + 11xy ).
    • Subtracting polynomials follows the same principles, such as in the problem with ( 6x + 3xy - 7y ) and ( 3x - 2xy + 6y ) resulting in ( -3x - 5xy + 13y ).

    Multiplication of Polynomials

    • Multiply each term in the first polynomial by each term in the second to obtain partial products.
    • Arrange like terms and simplify for the final expression.
    • For binomials, the FOIL (First, Outside, Inside, Last) method is applicable, ensuring an organized approach to multiplication.

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    Description

    This quiz covers the fundamental concepts of the Real Number System as outlined in Math030, focusing on sets and their properties. It introduces students to basic definitions and examples, essential for understanding algebraic principles. Prepare to test your knowledge on the foundational elements of sets and their applications in mathematics.

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