Questions and Answers
What is a combination of rational and irrational numbers?
Which of the following is a property of rational numbers?
What is the degree of the polynomial 3x^3 - 2x^2 + x - 1?
What is a polynomial with only one term called?
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How do you add or subtract polynomials?
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What is a pair of numbers (x, y) that represent a point on the Cartesian plane?
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What is an example of an irrational number?
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Study Notes
Number Systems
- Real Numbers: A combination of rational and irrational numbers.
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Rational Numbers: Can be expressed in the form p/q, where p and q are integers and q ≠ 0.
- Properties:
- Closure: The result of adding, subtracting, multiplying, or dividing two rational numbers is always a rational number.
- Commutativity: The order of rational numbers does not change the result of addition and multiplication.
- Associativity: The order in which rational numbers are added or multiplied does not change the result.
- Distributivity: a(b + c) = ab + ac
- Properties:
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Irrational Numbers: Cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.
- Examples: π, e, √2
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Recurring Decimals: Decimals that have a repeating pattern of digits.
- Example: 0.12341234...
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Non-Recurring Decimals: Decimals that do not have a repeating pattern of digits.
- Example: π
Polynomials
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Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
- Example: 2x^2 + 3x - 4
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Degree of a Polynomial: The highest power of the variable in the polynomial.
- Example: The degree of 2x^2 + 3x - 4 is 2.
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Terms of a Polynomial: The individual parts of the polynomial separated by + or - signs.
- Example: The terms of 2x^2 + 3x - 4 are 2x^2, 3x, and -4.
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Types of Polynomials:
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Monomial: A polynomial with only one term.
- Example: 2x^2
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Binomial: A polynomial with two terms.
- Example: x^2 + 3x
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Trinomial: A polynomial with three terms.
- Example: x^2 + 3x - 4
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Monomial: A polynomial with only one term.
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Addition and Subtraction of Polynomials:
- To add or subtract polynomials, combine like terms.
- Example: (2x^2 + 3x) + (x^2 - 2x) = 3x^2 + x
Coordinate Geometry
- Cartesian Plane: A two-dimensional plane with a horizontal x-axis and a vertical y-axis.
- Coordinates: A pair of numbers (x, y) that represent a point on the Cartesian plane.
- Plotting Points: To mark a point on the Cartesian plane using its coordinates.
- Distance Formula: The distance between two points (x1, y1) and (x2, y2) is given by √((x2 - x1)^2 + (y2 - y1)^2).
- Section Formula: The coordinates of a point that divides a line segment in the ratio m:n are given by ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)).
- Midpoint Formula: The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by ((x1 + x2) / 2, (y1 + y2) / 2).
Number Systems
- Real numbers combine rational and irrational numbers.
- Rational numbers can be expressed as p/q, where p and q are integers and q ≠ 0, and have properties of closure, commutativity, associativity, and distributivity.
- Irrational numbers cannot be expressed as p/q, and examples include π, e, and √2.
- Recurring decimals have a repeating pattern of digits, such as 0.12341234..., while non-recurring decimals do not, like π.
Polynomials
- A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.
- The degree of a polynomial is the highest power of the variable, such as 2 in 2x^2 + 3x - 4.
- Terms of a polynomial are individual parts separated by + or - signs, like 2x^2, 3x, and -4 in 2x^2 + 3x - 4.
- Types of polynomials include monomials (one term), binomials (two terms), and trinomials (three terms).
- To add or subtract polynomials, combine like terms, such as (2x^2 + 3x) + (x^2 - 2x) = 3x^2 + x.
Coordinate Geometry
- The Cartesian plane is a two-dimensional plane with a horizontal x-axis and a vertical y-axis.
- Coordinates are a pair of numbers (x, y) representing a point on the Cartesian plane.
- Plotting points involves marking a point on the plane using its coordinates.
- The distance formula calculates the distance between two points (x1, y1) and (x2, y2) as √((x2 - x1)^2 + (y2 - y1)^2).
- The section formula gives the coordinates of a point dividing a line segment in the ratio m:n as ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)).
- The midpoint formula calculates the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) as ((x1 + x2) / 2, (y1 + y2) / 2).
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Description
Explore the properties and definitions of real and rational numbers, including their properties and forms. Learn about closure, commutativity, and associativity of rational numbers.