Polynomial Quotes

I had a polynomial once. My doctor removed it.

Obligasyon ko bang pasanin ang mga problema ng math? Bakit? Bababa ba ng bill ko sa internet pag nag-factor ako ng quadratic trinomial? Malulutas ba ng Laws of Exponents ang problema natin sa basura? Mababawasan ba ng Associative Law for Multiplication ang mga krimen sa bansa? Makakabuti ba sa mag-asawa kung malalaman nila ang sum and difference of two cubes? Maganda ba sa sirkulasyon ng dugo ang parallelogram, polynomial at cotangent? Makatwiran bang pakisamahan ang mga irrational numbers? Anak ng scientific calculator!

I know this may sound like an excuse," he said. "But tensor functions in higher differential topology, as exemplified by application of the Gauss-Bonnett Theorem to Todd Polynomials, indicate that cohometric axial rotation in nonadiabatic thermal upwelling can, by random inference derived from translational equilibrium aggregates, array in obverse transitional order the thermodynamic characteristics of a transactional plasma undergoing negative entropy conversions." "Why don't you just shut up," said Hardesty.

Furious, the beast writhed and wriggled its iterated integrals beneath the King’s polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann’s Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out, fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier-—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, “Hurrah! Victory!!

So they rolled up their sleeves and sat down to experiment -- by simulation, that is mathematically and all on paper. And the mathematical models of King Krool and the beast did such fierce battle across the equation-covered table, that the constructors' pencils kept snapping. Furious, the beast writhed and wriggled its iterated integrals beneath the King's polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann's Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F_1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, "Hurrah! Victory!!

What this means is that the (Infinity) of points involved in continuity is greater than the (Infinity) of points comprised by any kind of discrete sequence, even an infinitely dense one. (2) Via his Diagonal Proof that c is greater than Aleph0, Cantor has succeeded in characterizing arithmetical continuity entirely in terms of order, sets, denumerability, etc. That is, he has characterized it 100% abstractly, without reference to time, motion, streets, noses, pies, or any other feature of the physical world-which is why Russell credits him with 'definitively solving' the deep problems behind the dichotomy. (3) The D.P. also explains, with respect to Dr. G.'s demonstration back in Section 2e, why there will always be more real numbers than red hankies. And it helps us understand why rational numbers ultimately take up 0 space on the Real Line, since it's obviously the irrational numbers that make the set of all reals nondenumerable. (4) An extension of Cantor's proof helps confirm J. Liouville's 1851 proof that there are an infinite number of transcendental irrationals in any interval on the Real Line. (This is pretty interesting. You'll recall from Section 3a FN 15 that of the two types of irrationals, transcendentals are the ones like pi and e that can't be the roots of integer-coefficient polynomials. Cantor's proof that the reals' (Infinity) outweighs the rationals' (Infinity) can be modified to show that it's actually the transcendental irrationals that are nondenumerable and that the set of all algebraic irrationals has the same cardinality as the rationals, which establishes that it's ultimately the transcendetnal-irrational-reals that account for the R.L.'s continuity.)

Certainly she can't and won't measure what is measureless, what neither terminates nor repeats, what is beyond even the transcendental of π - though HE doesn't think so - what is beyond polynomials and quadratic formulas, beyond the rational and irrational, the humanist and the logical, beyond the minds of the Cantors and the Dedekinds, the Renaissance philosophers and the Indian Tantrists, what falls instead into the realm of gods and kinds, of myth, of dawn of man, of the mystery of mankind - that there is a space inside her designed solely for him and despite clear Euclidian impossibilities not only does everything, in plenary excess, cleave like it's meant to, but it makes her feel what math cannot explain, what science cannot explain. What nothing can explain.

The Fundamental Theorem of Algebra If is a polynomial of degree , then has at least one zero in the complex number domain. In other words, there is at least one complex number such that .

For me, equation are nothing more than balancing numbers. The one who has systematic quicker way to expand power of number or polynomial , can solve almost all kind of equation in one line.

An interesting class of problems consists of those that demand a polynomial amount of space. This class is denoted as PSPACE. It is known that NP, the set of problems that can be solved in exponential or factorial time, can be solved in polynomial space. In other words, NP is a subset of PSPACE.

Alarmist rhetoric aside, the United States is not about to lose its primacy because students in Estonia are better at factoring polynomials. Other aspects of U.S. culture—a unique combination of creativity, entrepreneurship, optimism, and capital—have made it the most fertile ground in the world for innovation. That’s why bright kids from all around the globe dream of getting their green cards to work here.

How should you go about claiming your award money? There are two possible directions to take. You can try to prove that P = NP or you can aim to show that P is not equal to NP. To show that P = NP, all you have to do is take one of your favorite NP-Complete problems and find a polynomial algorithm that solves it. As we have seen, if you do find such an algorithm, then all NP problems will be solvable in a polynomial amount of operations. It might seem strange to think that a problem that demands an exponential or factorial amount of operations can be done in a polynomial amount of operations. It might seem strange to think that a problem that demands an exponential or factorial amount of operations can be done in a polynomial amount of operations. However, we saw something similar with the Euler Cycle Problem. Rather than look through all n! possible cycles to see if any are Euler cycles, we used the trick of checking if the number of edges touching each vertex is even or not. Does a similar trick for the Hamiltonian Cycle Problem exist? For many years, the smartest people around have been looking for such a trick or algorithm and have not been successful. However, you might possess some deeper insight that they lack. Get to it! On the other hand, you can try to show that P is not equal to NP. One way to do this is to take an NP problem and show that no polynomial algorithm exists for it. It so happens that it is very hard to prove such a claim: there are a lot of algorithms out there. This has turned out to be one of the hardest problems in mathematics. As a final hint, it should be noted that most researchers believe that P is not equal to NP.

In summary, all foreseeable future improvements to computer technology are essentially impotent in the face of these NP problems. The only way that these problems will be easily solved is to find nice polynomial algorithms for them. We will show in the next section why most researchers believe that there are no better algorithms for these problems. It looks as though they will remain problems that cannot be solved in a reasonable amount of time. These problems are not hard because we lack the technology to solve them. Rather they are hard because of the nature of the problems themselves. They are inherently hard and will probably remain on the outer limits of what we can solve.

There is a somewhat surprising source of cyclic groups: if p is prime, the group ((Z/pZ) ! , ·) is cyclic. We will prove a more general statement when we have accumulated more machinery (Theorem IV.6.10), but the adventurous reader can already enjoy a proof by working out Exercise 4.11. This is a relatively deep fact; note that, for example, (Z/12Z) ! is not cyclic (cf. Exercise 2.19, and Exercise 4.10). The fact that (Z/pZ) ! is cyclic for p prime means that there must be integers a such that every non-multiple of p is congruent to a power of a; the usual proofs of this fact are not constructive, that is, they do not explicitly produce an integer with this property. There is a very pretty connection between the order of an element of the cyclic group (Z/pZ) ! and the so-called ‘cyclotomic polynomials’; but that will have to wait for a little field theory (cf. Exercise VII.5.15

There's something about Algebra, I just can't figure it out Polynomials, derivatives, quadratic equations, I see no absolute value in them A bunch of irrational numbers With square roots and exponential functions I'm still trying to see through the horizontal and vertical blurred lines This all reminds me Y I left my X-

An algebraic integer of degree two is simply a root of a quadratic polynomial of the form X2 + aX + b with a, b ordinary integers.

Our brain has two halves: one is responsible for the multiplication of polynomials and languages, and the other half is responsible for orientation of figures in space and all the things important in real life. Mathematics is geometry when you have to use both halves.

If a problem can be solved in polynomial time we say it belongs to the complexity class P. So the problem that consists of multiplying two numbers together belongs to P. Suppose that instead of solving the problem, someone gives you the answer and you just have to check that the answer is correct. If this process of checking that an answer is correct takes polynomial time, then we say the problem belongs to complexity class NP.* The problem of factoring a large number into the product of two primes belongs to NP.

In particular, the three roots to a cubic polynomial either must all be real, or there must be one real root and one conjugate pair.

By now the reader is certainly convinced that group theory shows up in diverse situations. But it would be a great disservice to the history of mathematics if I did not mention one more application, the very reason that group theory was invented! In the nineteenth century, two young mathematical prodigies, Neils Abel and Evariste Galois, solved a mathematical problem that had stood unsolved for centuries. It has come to be called 'the unsolvability of the quintic.' It is one of the great discoveries in mathematics, and when you come to Chapter 10 of this book, you will be ready to read about it in some detail. It rests on the fact that the solutions to polynomial equations have a certain relationship to one another. They form a group.

The other key concept I shall refer to freely all through this book is that of a polynomial. The etymology of this word is a jumble of Greek and Latin, with the meaning “having many names,” where “names” is understood to mean “named parts. ” It seems to have first been used by the French mathematician François Viète in the late 16th century, showing up in English a hundred years later

In 1994, a mathematics professor at MIT, Peter Shor, developed a quantum algorithm to efficiently generate prime factors of large numbers, surpassing classical computer capabilities. Shor's algorithm represents a breakthrough in quantum computing, enabling the polynomial-time solution of prime factorization problems.