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

in agreement with Eq. (2.161)– which may be condensed to

(2.163)

where a 1, a 2, …, a n−1denote intermediate coefficients of Taylor’s expansion; one eventually obtains

(2.164)

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

(2.165)

upon straightforward algebraic simplification. If ξ = r 1is a root of P n{ x }, then

(2.166)

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

(2.167)

– so x − r 1may be factored out to produce

(2.168)

therefore, r 1being a root of P nimplies indeed that x − r 1is 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 − r 1, 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 n th‐ 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 b 1, b 2, …, b n−2denote 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 r 2of P n−1{ x }, abiding to

(2.176)

then Eq. (2.175)simplifies to

(2.177)

since x − r 2appears 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 n th‐degree polynomial holds n roots, i.e. r 1, r 2, …, 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 (− r 1) × (− r 2) × … × (− r n) of factors x − r 1, x − r 2, …, 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 − r 1, x − r 2, …, x − r n, respectively, in Eq. (2.182)– thus leading to the trivial result