A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

When n is an integer, the solution P < sub > n </ sub >( x ) that is regular at x = 1 is also regular at x = − 1, and the series for this solution terminates ( i. e. is a polynomial ).

When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as ( x − r )< sup > 3 </ sup >, with the same root r occurring three times, then the solution would take the form

# is a polynomial ( of degree ≤ n − 1 ) in .< ref > Proof: When A is normal, use Lagrange's interpolation formula to construct a polynomial P such that, where are the eigenvalues of A .</ ref >

When the ring is a field, the Cayley – Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.

When the entries of A are nonnegative, however, the permanent can be computed approximately in probabilistic polynomial time, up to an error of εM, where M is the value of the permanent and ε > 0 is arbitrary.

When, with Stirling's or Bessel's, the last term used includes the average of two differences, then one more point is being used than Newton's or other polynomial interpolations would use for the same polynomial degree.

When K is the real numbers, some of the p < sub > i </ sub > might be quadratic, so in the partial fraction decomposition a quotient of a linear polynomial by a power of a quadratic will occur.

When a is a non-positive integer, − n, is a polynomial.

When a is a non-positive integer, − n, is a polynomial.

When in a polynomial in of arbitrary degree the coefficient of the highest term is = 1, and all following coefficients are whole ( real, complex ) numbers, into which a certain ( real resp.

For univariate algebraic equations these are also called roots, even if, properly speaking, one should say the solutions of the algebraic equation P = 0 are the roots of the polynomial P. When solving an equation, it is important to specify in which set the solutions are allowed.

When at least t out of the n players reveal their points, there is sufficient information to fit a ( t-1 ) th degree polynomial to them, the first coefficient being the secret.

When using the term " quadratic polynomial ", authors sometimes mean " having degree exactly 2 ", and sometimes " having degree at most 2 ".

When the prime is 2, it is conventional to express elements of GF ( p < sup > n </ sup >) as binary numbers, with each term in a polynomial represented by one bit in the corresponding element's binary expression.

When an integer can be represented as a polynomial with small coefficients, the norms that arise are much smaller than those that arise when an integer is represented by a general polynomial.

When people say that the genome of a sexually reproducing species has been " sequenced ", typically they are referring to a determination of the sequences of one set of autosomes and one of each type of sex chromosome, which together represent both of the possible sexes.

When people are asked to make up a random-looking sequence of coin tosses, they tend to make sequences where the proportion of heads to tails stays closer to 0. 5 in any short segment than would be predicted by chance ( insensitivity to sample size ); Kahneman and Tversky interpret this to mean that people believe short sequences of random events should be representative of longer ones.

When the transposable elements lack additional genes, they are known as insertion sequences.

When injected into plants, these proteins can enter the nucleus of the plant cell, bind plant promoter sequences, and activate transcription of plant genes that aid in bacterial infection.

When hash functions and fingerprints are used to identify similar data, such as homologous DNA sequences or similar audio files, the functions are designed so as to maximize the probability of collision between distinct but similar data.

When working with stationary sources with memory, it is necessary to modify the definition of the rate distortion function and it must be understood in the sense of a limit taken over sequences of increasing lengths.

: When the sequences of proteins with the same function are examined in greater detail, some are found to share high sequence similarity.

; Large numbers of query sequences ( megablast ): When comparing large numbers of input sequences via the command-line BLAST, " megablast " is much faster than running BLAST multiple times.

; DNA mapping: When working with a known species, and looking to sequence a gene at an unknown location, BLAST can compare the chromosomal position of the sequence of interest, to relevant sequences in the database ( s ).

When the act appeared on his TV show, Southern CBS affiliates and racist Southern politicians complained of their participating in dance sequences with white women.

When used as a choreographic technique, movement sequences that emerge during Jam research or rehearsals may be adapted and set to form a part of set choreography, or a score ( a set of rules or limiting factors and transitions ) may be employed to give dancers a structure to navigate through a performance.

When R isn't commutative one needs the more careful statement, that ( if M is a left R-module ) the tensor product with M maps exact sequences of right R-modules to exact sequences of abelian groups.

When first unveiled in the concluding sequences of Star Trek IV, the main hull numbers are changed to NCC-1701-A, but the smaller hull numbers on the side of the lower part of the model still read NCC-1701, an oversight by the effects crew.

When PC games with full motion video ( FMV ) sequences were popular in the mid-to-late 1990s, PC Gamer's CD-ROM included elaborate FMV sequences featuring one of their editors.

When the action was indicated in a Republic script, it said " see Yakima Canutt for action sequences.

When people engage in these behavioral sequences they are dependent to some extent on their relational partner.

When R is a PID, then the correct statement of the Künneth theorem is that for any topological spaces X and Y there are natural short exact sequences

When questioned regarding difficulties in developing the game, Takahashi noted the effort used in making the opening sequences, developing the Special Games, and animations, which caused problems with meeting the schedule.

When examining peptides for synthesis and immunization, it is recommended that certain residues and sequences be avoided due to potential synthesis problems.

When he heard that Paul Whiteman was looking for singers to replace the Rhythm Boys, Mercer applied and got the job, `` not for my voice, I'm sure, but because I could write songs and material generally ''.

When asked whether the UK should replace Trident, without being told of the cost, 44 % of respondents said " Yes " and 46 % said " No ".

When this insulation is insufficient to maintain body temperature, they may resort to shivering -- rapid muscle contractions that quickly use up ATP, thus stimulating cellular metabolism to replace it and consequently produce more heat.

When dealing with three or fewer variables, it is common to replace x < sub > 1 </ sub > with just x, x < sub > 2 </ sub > with y, and x < sub > 3 </ sub > with z, as appropriate.

When Paschal died, these nobles made strenuous efforts to replace him with a candidate of their own ; and despite the fact that the clergy put forward a candidate likely to continue the policy of Paschal, the nobles were successful in their attempt.

When long-serving party leader Vernon Treatt announced his resignation in August 1954, Askin put his name forward to replace him.

When slang spreads beyond the group or subculture that originally used it, its original users often replace it with other, less-recognised terms to maintain group identity.

When it first launched the internal conflict in Peru in 1980, its stated goal was to replace what it saw as bourgeois democracy with " New Democracy ".

When the ball is not in play, the libero can replace any back-row player, without prior notice to the officials.

When the energy is available, arcjets are well suited to station keeping in orbit and can replace monopropellant rockets.

When he appointed Robert of Jumièges as Archbishop of Canterbury in 1051, he chose the leading craftsman Spearhafoc to replace Robert as Bishop of London.

When the United States government de-funded the NSFNET backbone, Internet exchange points were needed to replace its function, and initial governmental funding was used to aid the preexisting MAE and bootstrap three other exchanges, which they dubbed NAPs, or " Network Access Points ," in accordance with the terminology of the National Information Infrastructure document.

When Wim Jansen was appointed as the interim manager to replace Günder Bengtsson and Pim Verbeek after a 6 – 0 defeat against PSV, the outlook began to improve for the club.

When people began to fear that James's second wife, Mary, would produce a Catholic son and heir, a movement grew to replace him with his elder daughter Princess Mary and his son-in-law / nephew, William of Orange.

When Bavasi let Ryan leave after a 16 – 14 record in the 1979 season, Bavasi remarked he only needed to replace Ryan with two 8 – 7 pitchers, adding, " I think my plumber could do that.

When boards fail the test, technicians may desolder and replace failed components, a task known as rework.

When Thor discovers this, he grabs hold of Loki, resulting in Loki swearing to have a headpiece made of gold to replace Sif's locks.

When Frankfurter retired in 1962 and President John F. Kennedy named labor lawyer Arthur Goldberg to replace him, Warren finally had the fifth liberal vote for his majority.

When Charles Portal was made Chief of the Air Staff in October 1940 he supported Douglas, moving Park and Dowding and appointing Douglas to replace Dowding as Air Officer Commanding-in-Chief ( AOCinC ) of Fighter Command.

When Excelsior promoted to the Eredivisie after the season 2009 – 10, the club decided to replace the grass surface with artificial turf.

When the war broke out, James Pleasants, a native of the county, insisted he replace his uncle in the Goochland Light Dragoons ( known during the war as Co. F, 4th Virginia Cavalry ).

When Count Desnouettes died in 1822, Allen Glover was appointed to replace him.

When this church was demolished it was brought to Winschoten to replace an old altar.

When built, it will replace Jefferson High School and serve grades 9-12.

When residents rebuilt, they constructed substantial brick houses to replace many of the old wooden structures.