In Exercise 35.12, we considered polynomial division. In this problem, you are asked to develop the concept of polynomial gcd. Polynomials in this problem may be assumed to have rational coefficients. a. Let p and q be nonzero polynomials. Write a careful definition for common divisor and greatest common divisor of p and q. In this context, greatest refers to the degree of the polynomial. b. Show, by giving an example, that there need not be a unique gcd of two nonzero polynomials. c. Let d be a greatest common divisor of nonzero polynomials p and q. Prove that there exist polynomials a and b such that ap C bq D d. d. Give a careful definition of relatively prime for nonzero polynomials. e. Prove that two nonzero polynomials p and q are relatively prime if and only if there exist polynomials a and b such that ap C bq D 1. f. Let p D x 4 3x2 1 and q D x 2 C 1. Show that p and q are relatively prime by finding polynomials a and b such that ap C bq D 1.
Adequately Defined Variables Variable: A symbol used to represent a particular mathematical or logical entity with a particular value Variable Declaration: A statement stating that a certain symbol will be used to represent a particular mathematical or logical entity Adequately Defined Variables: Variables with unambiguously specified entities and values ◦ Entity Examples: Integer, Real Number, etc. Global vs Local Variables Global Definition: If the declaration of a variable’s entity and the setting of its value is preserved throughout an argument, unless its value is reset by a global redefinition Local Definition: If the declaration of a variable’s entity and the setting of its value is meaningful for only a limited section of an argument, after which the symbol loses the previous definition and must be redefined before it can meaningfully be used again ◦ Example: Ifthen statements ◦ Note: A variable is not adequately defined if it is only declared in an if then statement Congruence (mod n) – congruence: Let n be a positive integer, and suppose that a and b are any integers… ◦ “a is congruent to b modulo n (“a = b (mod n)”) if and only if a – b is an integer multiple of n” ◦ Theorem: “a = (a + n)(mod n) and a = (a – n)(mod n)” ◦ Corollary: “a = (a + bn)(mod n) Statements in Logic Statement: A sentence that is true or false, but not both