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    Basic Algebra

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    Roots and polynomials

    If z is an n-th term of 1, show taht 1+z+z^2+z^3+....+z^n-1=0. Solve the equation with n=5 and hence factorize 1+z+z^2+z^3+z^4 into linear factors with complex coefficient and then into quadratic factors with real coefficients.

    Introduction to Proofs Types

    Describe the steps in and/or define how to accomplish the following types of proofs: a. That two sets are equal. b. That two sets are disjoint. c. A proof by contra-positive. d. A proof by contradiction. e. A proof by Mathematical Induction. (Question is repeated in attachment)

    Equivalence Classes Relationships

    Question in attachment is as follows: Consider all lines in the plane. If a relationship between 2 lines is defined by the expression that their slopes are equal, prove that this relationship is an equivalence relationship. If we consider the set of all lines in the plane, how would you uniquely identify the equivalence clas

    Determining measurements using a quadratic equation.

    Reducing the Size of a candy Bar: A jumbo chocolate bar with a rectangular shape measures 12 centimeters in length, 7 centimeters in width, and 3 centimeters in thickness. Due to the escalating costs of cocoa, management decides to reduce the volume of the bar by 10%. To accomplish this reduction, management decides the new bar