Prerequisites Flashcards
Variable
A letter used to represent various numbers
Algebraic Expression
A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots
Exponential Notation
Evaluating an Algebraic Expression
To find the value of the expression for a given value of the variable
The Order of Operations Agreement
Equation
Formed when and equal sign is placed between two algebraic expressions
Formula
An equation that uses variables to express a relationship between two or more quantities
Mathematical Modeling
The process of finding formulas to describe real-world phenomena
Mathematical Model
Formulas, together with the meaning assigned to the variables
Model Breakdown
When a mathematical model gives an estimate that is not a good approximation or is extended to include values of teh variable that do not make sense
Set
A collection of objects whose contents can be clearly determined
Elements
The objects in a set
Roster Method
The braces, { }, indicate that we are representing a set. Uses commas to separate the elements of the set
Set-Builder Notation
The elements of the set are described but not listed.
{x | x is a counting number less than 6}
Read as: “x” –” | “ such that – “x is a counting number less than 6”
Intersection of Sets
Union of Sets
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers
The set of numbers that are either reational or irrational
{x | x is rational or irrational}
Real Number Line
A graph used to represent the set of real numbers
Origin
An arbitrary point labeled as 0 on a number line
Unit Distance
The distance from 0 to 1 on a number line
Positive Numbers
Numbers to the right of 0 on the real number line
Negative Numbers
Numbers to the left of 0 on the real number line
Graphed
On a real number line, placing a dot at the correct location for each number
Inequality Symbols
< and >
Absolute Value
Properties of Absolute Value
Distance between Two Points on the Real Number Line
If a and b are any two points on a real number line, then the distance between a and b is given by
a - b | or | b - a | , where | a - b | = | b - a |.
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication
Distributive Property of Multiplication over Addition
Identity Property of Addition
Identity Property of Multiplication
Inverse Property of Addition
Inverse Property of Multiplication
Subtraction
Division
Terms
The parts of an algebraic expression that are separated by addition
Coefficient
The numerical part of a term.
Example: 7x
The coefficient is: 7
Factors
The parts of each term that are multiplied.
The factors of the term 7x are 7 and x.
Like Terms
Terms that have exactly the same variable factors.
Example - 3x and 7x are like terms
Simplified
When the parentheses have been removed and like terms ahve been combined in an algebraic equation
Properties of Negatives
The Product Rule
The Quotient Rule
The Zero Exponent Rule
The Negative Exponent Rule
Negative Exponents in Numerators and Denominators
The Power Rule (Powers to Powers)
Product to Powers
Quotients to Powers
Simplifying Exponential Expressions
Scientific Notation
Converting from Decimal to Scientific Notation
Principal Square Root
Simplifying the square root of a squared term
The Product Rule for Square Roots
The Quotient Rule for Square Roots
Multiplying Conjugates
Principal nth Root of a Real Number
Finding nth Roots of Perfect nth Powers
The Product and Quotient Rules for nth Roots
Definition of a^(1/n)
Definition of a^(m/n)
Polynomial
A single term or the sum of two or more terms containing vairalbes with whole-number exponents
Standard form of a Polynomial
Writing the terms of a polynomial in the order of descending powers fo the variable
The Degree of axn
Monomial
A simplified polynomial that has exactly one term
Binomial
A simplified polynomial that has two terms
Trinomial
A simplified polynomial with three terms
Degree of a Polynomial
The greatest degree of all terms of the polynomial.
Example - 4x2 + 3x is a binomial with a degree of 2
Definition of a Polynomial in x
Multiplying Polynomials When Neither Is a Monomial
Multiply each term of one polynomial by each term of the other polynomial. Then combine like terms.
FOIL Method
Used when multiplying two binomials. Represents the order of First, Outer, Inner, Last.
The Product of the Sum and Difference of Two Terms
The Square of a Binomial Sum
The Square of a Binomial Difference
Sum and Difference of Two Terms
Squaring a Binomial
Cubing a Binomial
Greatest Common Factor (GCF)
An expression of the highest degree that divides each term of the polynomial. The distributive property in reverse direction
ab + ac = a(b + c)
Factoring Trinomials
The Difference of Two Squares
Factoring Perfect Square Trinomials
Factoring the Sum or Difference of Two Cubes
A Strategy for Factoring a Polynomial
Rational Expression
The quotient of two polynomials
Example - ( x - 2 ) / 4, 4 / ( x - 2 ), x / ( x2 - 1 )
Simplifying Rational Expressions
- Factor the numerator and the denominator completely.
- Divide both the numerator and the denominator byany common factors.
Multiplying Rational Expressions
- Factor all numerators and denominators completely.
- Divide numerators and denominators by common factors.
- Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators.
Finding the Least Common Denominator
- Factor each denominator completely.
- List the factors of the first denominator.
- Add to the list in step 2 any factors of the second denominator that do not appear in the list.
- Form the product of each different factor fom the list in step 3. This product is the least common denominator.
Adding and Subtracting Rational Expressions That Have Different Denominators
- Find the LCD of the rational expressions.
- Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
- Add or subtract numerators, placing the resulting expression over the LCD.
- If possible, simplify the resulting rational expression.
