MATH 1111 Section P.1 Bland Variable A letter or symbol used to represent a number. Algebraic Expressions - An algebraic expression is a combination of variables and numbers using operations. Coefficient A number being directly multiplied by a variable. Ex: In the algebraic expression coefficient of x., 6 is the ( ) Order of Operations When we have multiple operations we must follow a particular order. *Note: Some calculators know the order of operations and some do not. Know how to use your calculator! Evaluating Algebraic Expressions means to find the value of the expression for a given value of the variable. Example: Evaluate for. Equation - An equation is formed when an equal sign is place between two algebraic expressions. Finding formulas to represent real world phenomena is called mathematical modeling.
MATH 1111 Section P.1 Bland Sets A set is a collection of objects whose contents can be clearly determined. The objects in a set are called the elements of the set. Example: The set of numbers used for counting can be represented by The braces, { }, indicate that we are representing a set. This form of representation, is called the roster method, it uses commas to separate the elements of the set. The three dots at the end are called ellipsis, indicating the list goes on forever. Another way to denote a set is to use set builder notation. In this notation, the elements of the set are described, but not listed. Here is an example:. The same set written using the roster method is. Example: Find the intersection: Example: Find the intersection: {2, 4, 6} {3, 5, 7}. Example: Find the union: {7, 8, 9, 10, 11} {6, 8, 10, 12}.
MATH 1111 Section P.1 Bland Absolute Value is the distance from a number to zero. For example, because is 3 places away from 0. because 5 is 5 places from 0.
MATH 1111 Section P.2 Bland 1. 2. 3. 4.
MATH 1111 Section P.2 Bland ( )
MATH 1111 Section P.3 Bland Principal Square Root If is a nonnegative real number, the nonnegative number such that, denoted by is the principal square root of. Simplifying For any real number,. In words, the principal square root of is the absolute value of. The Product Rule for Square Roots If and represent nonnegative real numbers, then and. The square root of a product is the product of the square roots. a) b) The Quotient Rule for Square Roots If and represent nonnegative real numbers and, then and The square root of a quotient is the quotient of the square roots. Example: Simplify using the quotient rule for square roots. Assume that x>0.
MATH 1111 Section P.3 Bland Property of Like Radicals (Terms) Two or more square roots can combined using the distributive property provided that they have the same radicand. Such radicals are called like radicals. a) b) Rationalizing Rationalizing a denominator involves rewriting a radical expression as an equivalent expression in which the denominator no longer contains any radicals. a) b) Definition of the Principal nth Root of a Real Number means that. If, the index, is even, then is nonnegative ( ) and is also nonnegative. If is odd, and can be any real numbers. a) b) c) The Definition of If represents a real number and is a positive rational number,, then Also, ( ).. Furthermore, if is a nonzero real number, then a) b) c)
MATH 1111 Section P.4 Bland The Degree of If, the degree of is. The degree of a nonzero constant is 0. The constant 0 has no defined degree. Example: Determine the degree of each term. Definition of a Polynomial in A polynomial in is an algebraic expression of the form where,,,,, and are real numbers,, and is a nonnegative integer. The polynomial is of degree, is the leading coefficient, and is the constant term. Example:, Adding/Subtracting Polynomials Combine like terms. Example: Simplify (Do Not Factor) a) Multiplying Polynomials Distribute terms. a) b) c) d) e) f)
MATH 1111 Section P.5 Bland Strategy used in MATH 0090 Given, multiply and and list all factors. Add all factors until you find. Split using the two factors and factor by grouping. Example:
MATH 1111 Section P.5 Bland STEP 1 Factor out the GCF. STEP 2 Check for special forms. Difference of two squares: Sum of two cubes: Difference of two cubes: STEP 3 Factor by grouping. To use this method we need there to be four terms. However, having four terms does not guarantee this method will work. STEP 4 Use the ac-method. To use this method the trinomial need to be in the form.
MATH 1111 Section P.6 Bland Rational Expressions are simply fractions with polynomials as the numerators and denominators. Which means it s a polynomial divided by a polynomial. Domain The domain is the possible numbers that can represent. For example has a domain that covers all the real numbers ( ). Find all the numbers that must be excluded from the domain of each rational expression: a) b) Simplifying rational expressions Simplify: a) b) d) e) f)