Algebra Chapter 1: Polynomials and Sets

Chapter 1 Polynomials

• Objectives:
  • Recall fundamental operations for signed numbers
  • Apply basic rules and concepts of rational expressions
  • Understand basic polynomial operations
• Key Concepts:
  • Algebra is a branch of mathematics that uses mathematical statements to describe relationships between varying things (e.g., supply and price).
  • Variables are used to represent quantities that vary.
  • Algebraic expressions combine letters and symbols to represent values and relationships.
  • Set: A collection of unique objects, elements.

The Real Number System

• Set: A collection of unique objects.
• Element: A unique object within a set.
• Set representation: Roster form and set-builder form.
  • Roster form lists elements.
  • Set-builder form describes elements using a rule.
• Examples:
  • Set of positive numbers
  • Set of counting numbers less than ten
  • Set of vowels in the English alphabet

Types of Numbers

• Natural numbers: Used for counting (1, 2, 3, etc.). Prime numbers have only 1 and themselves as factors; composite numbers have other factors besides 1 and themselves.
• Whole numbers: Natural numbers and zero.
• Integers: Whole numbers and their opposites.
• Rational numbers: Numbers that can be expressed as a fraction (ratio of two integers). Can be terminating or repeating decimals.
• Irrational numbers: Numbers that cannot be expressed as a fraction. Have non-terminating, non-repeating decimals (e.g., √2, π).
• Real numbers: All rational and irrational numbers.

Operations of Signed Numbers

• Adding integers with same sign: Add absolute values, keep the common sign.
• Adding integers with different signs: Subtract absolute values, use sign of larger absolute value.
• Subtracting integers: Change sign of subtrahend, then add.
• Multiplying integers: Like signs give positive result, unlike signs give negative result.
• Dividing integers: Like signs give positive result, unlike signs give negative result.

Algebraic Expressions

• Definition: A meaningful collection of numbers, variables, and signs of mathematical operations.
• Example: 8x², where 8 is the numerical coefficient, x is the variable, and 2 is the exponent (power).
• Laws of Exponents
  • Product of powers: am * an = am+n
  • Power of a power: (am)n= amn
  • Power of a product: (ab)m = ambm

Polynomials

• Definition: A special type of algebraic expression; terms are in the form axⁿ or bxyⁿ where a and b are real numbers and m and n are whole numbers.
• Examples: 2x, x² + 3x - 1, √2x² + 3x³y-x²y²-ху³+y⁴
• Monomial: One term (e.g., 3x²)
• Binomial: Two terms (e.g., 2x - 3y)
• Trinomial: Three terms (e.g., x² - 8x + 12)

Operations on Polynomials

• Addition/Subtraction: Combine like terms.
• Multiplication: Use distributive law.
• Division: Apply exponent rules, including quotients.

Special Products (Chapter II)

• Product of two binomials: The product of two binomial factors (e.g., (ax + b)(cx + d)) can be obtained by multiplying - first terms (of the binomials) - outside terms (of the binomials) - inside terms (of the binomials) - last terms (of the binomials)
• Product of the sum and difference of two numbers: (x +y)(x - y) = x² - y²
• Square of a binomial: (x + y)² = x² + 2xy + y² and (x - y)² = x² - 2xy + y²

Factoring (Chapter III)

• Removal of Common Factor: Divide the polynomial by the highest common factor.
• Difference of Two Squares: (x² - y²) = (x + y)(x - y)
• Perfect Trinomial Square: x² + 2xy + y² = (x+y)² and x² -2xy + y² = (x-y)²
• Trinomials of the form ax² + bx + c (where |a|>1):
  • Often solved by trial and error, finding factors that multiply to 'a' and 'c'. Their sum must be the value of 'b'.

Radicals (Chapter IV)

• Radicand: The number under the radical symbol.
• Index: The number outside the radical symbol (e.g. the 3 in 3√2).
• Properties of Radicals: Manipulating radicals according to their index and radicand.
• Operations on Radicals: Combining like radicals; multiplying radicals; dividing radicals.

Linear Equations (Chapter V)

• Equation: A mathematical statement that two algebraic expressions are equal.
• Linear Equation: An equation where the greatest exponent of the variable(s) is one.
• Conditional Equation: An equation is true for specific values of the variable, but not all values.
• Identity Equation: An equation that is true for all permissible values of the variables involved.
• Solving Linear Equations: Steps to find the unknown value in an equation.

This quiz explores fundamental concepts in Algebra, focusing on polynomials and the real number system. It covers operations with signed numbers, rational expressions, and basic polynomial operations, as well as understanding sets and their representations. Test your knowledge and grasp of these essential algebraic principles.