Divide 2x3 - 3x2 - 5x - 12 by x
... Suppose f(x) is divided by x - k using synthetic division. 1. If k≥ 0 and every number in the last line is nonnegative (pos. or zero), then k is an upper bound for the real zeros of f. 2. If k ≤ 0 and the numbers in the last line alternate nonnegative and nonpositive, then k is a lower bound for the ...
... Suppose f(x) is divided by x - k using synthetic division. 1. If k≥ 0 and every number in the last line is nonnegative (pos. or zero), then k is an upper bound for the real zeros of f. 2. If k ≤ 0 and the numbers in the last line alternate nonnegative and nonpositive, then k is a lower bound for the ...
Note on a conjecture of PDTA Elliott
... From these two results we get p \ (ar — bs) = c which contradicts (ab, c) = 1. Thus we can apply Ricci's theorem to obtain that there is a constant K = = K(a, b) and an infinitely many natural numbers p and q with at most K prime factors such that p = ft(x) and q = f2(x). From p = bx + r and g = dx ...
... From these two results we get p \ (ar — bs) = c which contradicts (ab, c) = 1. Thus we can apply Ricci's theorem to obtain that there is a constant K = = K(a, b) and an infinitely many natural numbers p and q with at most K prime factors such that p = ft(x) and q = f2(x). From p = bx + r and g = dx ...
CHAPTER 5. Convergence of Random Variables
... One of the most important parts of probability theory concerns the behavior of sequences of random variables. This part of probability is often called “large sample theory” or “limit theory” or “asymptotic theory.” This material is extremely important for statistical inference. The basic question is ...
... One of the most important parts of probability theory concerns the behavior of sequences of random variables. This part of probability is often called “large sample theory” or “limit theory” or “asymptotic theory.” This material is extremely important for statistical inference. The basic question is ...
Introduction to Discrete Mathematics
... Proof by Appel and Haken 1976; careful case analysis performed by computer; Proof reduced the infinitude of possible maps to 1,936 reducible configurations (later reduced to 1,476) which had to be checked one by one by computer. The computer program ran for hundreds of hours. The first significant c ...
... Proof by Appel and Haken 1976; careful case analysis performed by computer; Proof reduced the infinitude of possible maps to 1,936 reducible configurations (later reduced to 1,476) which had to be checked one by one by computer. The computer program ran for hundreds of hours. The first significant c ...
Understanding the Central Limit Theorem
... Design a logo for this project. Insert this logo in the forms created above. Pick a background color and a font color for the forms created. Include the following in the forms created: record navigation command buttons, record operations command buttons, and form operations ...
... Design a logo for this project. Insert this logo in the forms created above. Pick a background color and a font color for the forms created. Include the following in the forms created: record navigation command buttons, record operations command buttons, and form operations ...
Converse of the Pythagorean Theorem
... Babylonians, Egyptians, and Chinese were aware of this relationship before its discovery by Pythagoras. ...
... Babylonians, Egyptians, and Chinese were aware of this relationship before its discovery by Pythagoras. ...
Bloom`s Taxonomy applied to understanding the Pythagorean
... 7. Construct a situation in which the Pythagorean theorem would be used to solve a mathematical problem that musicians may have. Evaluation 1. The Pythagorean Theorem has been proved in over 100 different ways. What is the value in examining more than one proof of a Theorem? 2. How could you effecti ...
... 7. Construct a situation in which the Pythagorean theorem would be used to solve a mathematical problem that musicians may have. Evaluation 1. The Pythagorean Theorem has been proved in over 100 different ways. What is the value in examining more than one proof of a Theorem? 2. How could you effecti ...
Natural deduction for predicate logic
... and b that satisfy the statement of the theorem. An intuitionist would reject our previous proof of the theorem. This is not equivalent to rejecting the theorem itself. The result may actually possess an intuitionistically valid proof and therefore be perfectly acceptable. But such a proof must tell ...
... and b that satisfy the statement of the theorem. An intuitionist would reject our previous proof of the theorem. This is not equivalent to rejecting the theorem itself. The result may actually possess an intuitionistically valid proof and therefore be perfectly acceptable. But such a proof must tell ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.