9 Questions
What is the definition of a polynomial?
An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
What is the term for a polynomial with only one term?
Monomial
What is the result of adding 2x^2 + 3x + 1 and x^2 - 2x?
3x^2 + x + 1
What is the term for a polynomial with three terms?
Trinomial
What is the set of numbers that includes all rational and irrational numbers?
Real Numbers
What is the property of real numbers that states the order of numbers does not change the result?
Commutative Property
What is the result of multiplying (x + 2) and (x + 3)?
x^2 + 5x + 6
What is the set of numbers that includes only positive integers?
Natural Numbers
What is the property of real numbers that states the order in which numbers are grouped does not change the result?
Associative Property
Study Notes
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.
Learn about polynomials, including their definition, and the different types such as monomials, binomials, and trinomials.
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