Podcast Beta
Questions and Answers
What is the set of numbers that includes all rational and irrational numbers?
Which of the following is an example of an irrational number?
What is the property of real numbers that states the product of a real number and the sum of two real numbers is equal to the sum of the products of the real number with each of the two real numbers?
What is a polynomial with one term called?
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What is the result of adding two polynomials with like terms?
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What operation is used to multiply two polynomials?
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What is the result of expanding the expression (2x + 3) × (4x - 2)?
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What is the graphical method of solving a pair of linear equations in two variables?
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What does it mean if a pair of linear equations is dependent?
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What is the first step in the elimination method of solving a pair of linear equations in two variables?
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What is the definition of a pair of linear equations in two variables?
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Study Notes
Real Numbers
- Definition: Real numbers are a set of numbers that include all rational and irrational numbers.
- Rational Numbers: A rational number is a number that can be expressed as the quotient or fraction of two integers, e.g. 3/4, 22/7.
- Irrational Numbers: An irrational number is a number that cannot be expressed as a simple fraction, e.g. π, e, √2.
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Properties of Real Numbers:
- Closure property: The sum and product of two real numbers is always a real number.
- Commutative property: The order of real numbers does not change the result of addition and multiplication.
- Associative property: The order in which real numbers are added or multiplied does not change the result.
- Distributive property: The product of a real number and the sum of two real numbers is equal to the sum of the products of the real number with each of the two real numbers.
Polynomials
- Definition: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
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Types of Polynomials:
- Monomials: Polynomials with one term, e.g. 2x, 3y.
- Binomials: Polynomials with two terms, e.g. 2x + 3, x^2 + 4.
- Trinomials: Polynomials with three terms, e.g. x^2 + 2x + 1.
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Operations on Polynomials:
- Addition: Combine like terms, e.g. (2x + 3) + (4x - 2) = 6x + 1.
- Subtraction: Combine like terms, e.g. (2x + 3) - (4x - 2) = -2x + 5.
- Multiplication: Use the distributive property, e.g. (2x + 3) × (4x - 2) = 8x^2 - 4x + 12.
Pair of Linear Equations in Two Variables
- Definition: A pair of linear equations in two variables is a system of two equations with two variables, e.g. 2x + 3y = 7, x - 2y = -3.
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Methods of Solving:
- Substitution Method: Solve one equation for one variable, substitute into the other equation, and solve for the other variable.
- Elimination Method: Make the coefficients of one variable equal, add or subtract the equations to eliminate one variable, and solve for the other variable.
- Graphical Method: Graph the two equations on a coordinate plane, and find the point of intersection, which is the solution.
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Consistency of Linear Equations:
- Consistent: The pair of linear equations has a unique solution.
- Inconsistent: The pair of linear equations has no solution.
- Dependent: The pair of linear equations has infinitely many solutions.
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Description
Test your understanding of real numbers, polynomials, and linear equations in two variables. This quiz covers definitions, properties, and operations on these algebraic concepts.
