Algebra: Polynomials

Polynomials

Definition

• A polynomial is an expression consisting of variables (such as x or y) and coefficients (such as numbers) combined using only addition, subtraction, and multiplication.
• Polynomials can be thought of as simplified algebraic expressions.

Types of Polynomials

• Monomials: Polynomials with only one term (e.g. 3x^2)
• Binomials: Polynomials with two terms (e.g. x^2 + 3x)
• Trinomials: Polynomials with three terms (e.g. x^2 + 3x + 2)

Operations with Polynomials

• Addition and Subtraction: Combine like terms (e.g. 2x^2 + 3x + 1 + x^2 - 2x = 3x^2 + x + 1)
• Multiplication: Distribute each term in one polynomial to each term in the other (e.g. (x + 2)(x + 3) = x^2 + 5x + 6)

Number System

Types of Numbers

• Natural Numbers: Positive integers (e.g. 1, 2, 3, ...)
• Whole Numbers: Non-negative integers (e.g. 0, 1, 2, ...)
• Integers: Positive and negative whole numbers (e.g. ..., -3, -2, -1, 0, 1, 2, 3, ...)
• Rational Numbers: Fractions (e.g. 1/2, 3/4, 22/7)
• Irrational Numbers: Non-repeating, non-terminating decimals (e.g. π, e, √2)
• Real Numbers: Includes rational and irrational numbers

Properties of Real Numbers

• Commutative Property: The order of numbers does not change the result (e.g. a + b = b + a)
• Associative Property: The order in which numbers are grouped does not change the result (e.g. (a + b) + c = a + (b + c))
• Distributive Property: Multiplication distributes over addition (e.g. a(b + c) = ab + ac)

Polynomials

Definition

• A polynomial is an expression combining variables and coefficients using addition, subtraction, and multiplication.

Types of Polynomials

• A monomial is a polynomial with one term, e.g. 3x^2.
• A binomial is a polynomial with two terms, e.g. x^2 + 3x.
• A trinomial is a polynomial with three terms, e.g. x^2 + 3x + 2.

Operations with Polynomials

• To add or subtract polynomials, combine like terms, e.g. 2x^2 + 3x + 1 + x^2 - 2x = 3x^2 + x + 1.
• To multiply polynomials, distribute each term in one polynomial to each term in the other, e.g. (x + 2)(x + 3) = x^2 + 5x + 6.

Number System

Types of Numbers

• Natural numbers are positive integers, e.g. 1, 2, 3, ...
• Whole numbers are non-negative integers, e.g. 0, 1, 2, ...
• Integers include positive and negative whole numbers, e.g. ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers are fractions, e.g. 1/2, 3/4, 22/7.
• Irrational numbers are non-repeating, non-terminating decimals, e.g. π, e, √2.
• Real numbers include rational and irrational numbers.

Properties of Real Numbers

• The commutative property states that the order of numbers does not change the result, e.g. a + b = b + a.
• The associative property states that the order in which numbers are grouped does not change the result, e.g. (a + b) + c = a + (b + c).
• The distributive property states that multiplication distributes over addition, e.g. a(b + c) = ab + ac.