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

expansion; one eventually obtains

      (2.164)

after merging Eqs. (2.161) and (2.163), or else

(2.165)

      upon straightforward algebraic simplification. If ξ = r₁ is a root of P_n {x}, then

      (2.166)

by definition; hence, Eq. (2.165) degenerates to

      (2.167)

      – so x − r₁ may be factored out to produce

(2.168)

therefore, r₁ being a root of P_n implies indeed that x − r₁ is a factor of P_n – in full agreement with Eq. (2.159).

Inspection of Eq. (2.168) indicates that an (n − 1)th degree polynomial has been produced, viz.

(2.169)

      which may be reformulated to

      (2.170)

      – obtained after applying Newton’s binomial formula (to be derived shortly) in expansion of all powers of x − r₁, and then lumping terms associated with the same power of x so as to produce coefficients b_j; one may again proceed to Taylor’s expansion of P_n−1{x} as

(2.171)

in parallel to Eq. (2.161), since nth‐ and higher‐order derivatives of P_n−1{x} are nil. The resulting coefficients in Eq. (2.171) read

      (2.172)

      or, equivalently,

(2.173)

here b₁, b₂, …, b_n−2 denote intermediate coefficients of Taylor’s expansion. Equation (2.173) may then be taken advantage of to rewrite Eq. (2.171) as

      (2.174)

      where cancelation of (n − 1)! between numerator and denominator of the last term unfolds

(2.175)

      If one sets ξ equal to a root r₂ of P_n−1{x}, abiding to

      (2.176)

then Eq. (2.175) simplifies to

(2.177)

since x − r₂ appears in all terms of Eq. (2.177), it may be factored out to yield

(2.178)

– so insertion of Eq. (2.178) transforms Eqs. (2.168) and (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)