Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier Malcata

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rel="nofollow" href="#ulink_4cd274a5-a61a-5f4e-9a86-12c4b14b1002">(2.169) to

(2.179)

      as long as

      (2.180)

      The above process may be iterated as many times as the number of roots – knowing that an nth‐degree polynomial holds n roots, i.e. r₁, r₂, …, r_n (even though some of them may coincide); the final polynomial will accordingly look like

      (2.181)

      – in line with Eqs. (2.165) and (2.175), and consequently

(2.182)

will appear as general factorized form of any given polynomial, as suggested by Eqs. (2.168) and (2.179).

One important consequence of the extended product form of Eq. (2.182) is that the coefficients of each power in the original polynomial P_n {x} bear a direct relationship to its roots, according to

(2.183)

      generated with the aid of Eq. (2.135). For instance, the independent term of P_n {x} must result from the product of only (−r₁) × (−r₂) × … × (−r_n) of factors x − r₁, x − r₂, …, x − r_n, respectively, in Eq. (2.182), so one may state

      (2.184)

      similarly, one finds that the highest order term in Eq. (2.135) will necessarily result from the product of only x × x × ⋯ × x of factors x − r₁, x − r₂, …, x − r_n, respectively, in Eq. (2.182) – thus leading to the trivial result

      (2.185)

      By the same token, the terms in x_n−1 of Eq. (2.135) are accounted for by the product of x in x − x₁, x − x₂, …, x − x_i, x − x_i+1, …, x − x_n, respectively, of Eq. (2.182) by –r_i in x − r_i, thus generating ; this is then to be extended, via addition, to i = 1, 2, …, n, thus eventually giving rise to

      (2.186)

as apparent in Eq. (2.183). The coefficients of all remaining terms can be generated in a similar fashion; for power x^n−j in general, the x’s of n − j factors of the x − r_i form are to be picked up, and multiplied by the remaining j roots of the r_i form – and the results added up, thus unfolding a composite set of j summations that account for all such combinations. Special care is to be exercised to increment the lower limit of each summation by one unit relative to the lower limit of the previous summation, and decrement the upper limit of each summation by one unit relative to the upper limit of the next summation; in this way, each possible combination is counted just once (as it should).

      2.2.4 Splitting

      Once in possession of the equivalent result conveyed by Eq. (2.182) but applied to P_m {x}, one may revisit Eq. (2.141) as

(2.187)

      – or, after lumping constant b_m with the corresponding polynomial in numerator,

(2.188)

The roots r_k of the polynomial in denominator may, in general, take real or complex values (i.e. of the form α + ιβ); when s_l roots are equal to r_k, one may lump the corresponding binomials as instead of multiplying x − r_l by itself s_l times, i.e. Eq. (2.188) may alternatively appear as

(2.189)

as long as the r_l ’s represent the s distinct roots (or poles) of P_m {x}, each with multiplicity s_l, and . After splitting the denominator of the proper rational fraction in Eq. (2.189), one obtains

(2.190)

      – where s₁ was arbitrarily chosen among the (multiple) roots, and denotes an (m − s₁)th degree polynomial defined as

      (2.191)

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