Chapter 4
... 1. Upper Bound. If r > 0 and all numbers in the quotient row of the synthetic division, including the remainder, are nonnegative, then r is an upper bound of the real zeros of P(x). 2. Lower Bound. If r < 0 and all numbers in the quotient row of the synthetic division, including the remainder, alter ...
... 1. Upper Bound. If r > 0 and all numbers in the quotient row of the synthetic division, including the remainder, are nonnegative, then r is an upper bound of the real zeros of P(x). 2. Lower Bound. If r < 0 and all numbers in the quotient row of the synthetic division, including the remainder, alter ...
Full text
... "canonical" initial values, cf [4]) which seem to be more suitable for defining digital expansions and yield an index translation by 2. In [3] we have proved that the first conjecture is true in a much more general situation, i.e., for digital expansions with respect to linear recurrences with nonin ...
... "canonical" initial values, cf [4]) which seem to be more suitable for defining digital expansions and yield an index translation by 2. In [3] we have proved that the first conjecture is true in a much more general situation, i.e., for digital expansions with respect to linear recurrences with nonin ...
MEI Conference 2009 Proof
... 12. Every positive integer can be written in the form a 2 + b 2 − c 2 where a, b and c are integers 13. For any polynomial equation x n + an −1 x n −1 + ... + a2 x 2 + a1 + a0 = 0 where all the coefficients are integers, if any roots are rational numbers then they must be integers. 14. An equilatera ...
... 12. Every positive integer can be written in the form a 2 + b 2 − c 2 where a, b and c are integers 13. For any polynomial equation x n + an −1 x n −1 + ... + a2 x 2 + a1 + a0 = 0 where all the coefficients are integers, if any roots are rational numbers then they must be integers. 14. An equilatera ...
The Real Number Line and Types of Real Numbers
... Place each of the following numbers into the sets (boxes) to which they belong. Note that most of the numbers fit into more than one set. ...
... Place each of the following numbers into the sets (boxes) to which they belong. Note that most of the numbers fit into more than one set. ...
Polygonal Numbers
... very different that the square or triangular numbers. Polygonal numbers were first studied by the ancient Greeks (the Pythagoreans), who made several important observations that were later proved by more modern mathematicians. For instance, they observed that the common difference in the sequence of ...
... very different that the square or triangular numbers. Polygonal numbers were first studied by the ancient Greeks (the Pythagoreans), who made several important observations that were later proved by more modern mathematicians. For instance, they observed that the common difference in the sequence of ...
![[Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments (§2.3](http://s1.studyres.com/store/data/014954007_1-d36c768aba23f0b4aa633cb9a2a27ee2-300x300.png)
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... 2: Let IT be a partition whose Ferrers graph is embedded in the fourth quadrant. Each node (i, j) of the fourth quadrant which is not in the Ferrers graph of IT is said to possess an anti-hook difference p^ - kj relative to ir, where p^ is the number of nodes on the i t h row of the fourth quadrant ...
... 2: Let IT be a partition whose Ferrers graph is embedded in the fourth quadrant. Each node (i, j) of the fourth quadrant which is not in the Ferrers graph of IT is said to possess an anti-hook difference p^ - kj relative to ir, where p^ is the number of nodes on the i t h row of the fourth quadrant ...
Problems - My E-town - Elizabethtown College
... cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube? 2. (2005B1) Find a nonzero polynomial P (x, y) such that P (bac, b2ac) = 0 for all real numbers a. (Note: bνc is the greatest integer less than or equal to ν.) 3. (20 ...
... cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube? 2. (2005B1) Find a nonzero polynomial P (x, y) such that P (bac, b2ac) = 0 for all real numbers a. (Note: bνc is the greatest integer less than or equal to ν.) 3. (20 ...
A relation between partitions and the number of divisors
... Fix an m = 1, ..., n. We shall show that the coefficient αm for X m in Pn (X) equals d(m) − p(m). First, we determine the contributions from the separate quotients of P n (X) to αm . Fix an i = 0, ..., n − 1, and replace the denominator 1/(1 − X n−i ) in the ith quotient of Pn (X) by its power serie ...
... Fix an m = 1, ..., n. We shall show that the coefficient αm for X m in Pn (X) equals d(m) − p(m). First, we determine the contributions from the separate quotients of P n (X) to αm . Fix an i = 0, ..., n − 1, and replace the denominator 1/(1 − X n−i ) in the ith quotient of Pn (X) by its power serie ...