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

if the theorems on limits are blindly applied. To circumvent the unknown quantities of the 0/0 type, one may independently differentiate, with regard to k 2, numerator and denominator of either term in the right‐hand side of Eq. (2.128), according to

(2.129)

where minus signs may be dropped from numerator and denominator, factored out, and classical theorems on limits considered to reach

(2.130)

upon division of both numerator and denominator of the last term by 1 − k 2, one gets

(2.131)

or, equivalently,

(2.132)

once n + 1 has been factored out – which retrieves Eq. (2.83)pertaining to a plain arithmetic series, as expected. This conclusion can also be drawn graphically, by comparing the curves labeled as k 2 = 1 in Fig. 2.6with the lines in Fig. 2.4corresponding to the same ; and goes along with realization that

(2.133)

based on Eq. (2.110)and matching Eq. (2.76).

A polynomial in x refers to a sum of powers of integer exponent on x ; such algebraic functions are accordingly the easiest in terms of calculation – including numerical methods, as they basically involve just multiplications and additions. In principle, any continuous function of practical interest may be expressed via a polynomial (as will be derived later); hence, algebraic operations involving polynomials are of the utmost interest for process engineering. Besides the trivial operation of addition – where terms with identical exponents of x are merely lumped, i.e.

(2.134)

with

(2.135)

and

(2.136)

multiplication and division of polynomials appear as germane. An iterated version of the former entails the power of a polynomial, whereas the iterated version of the latter supports factorization of a polynomial – i.e. conversion from a sum to a product of simpler (usually linear) polynomials.

Recalling the two polynomials labeled as Eqs. (2.135)and (2.136)– of n th and m th degree, respectively, one may define their product P n P mas an ( n + m )th degree polynomial, viz.

(2.137)

upon application of the distributive property of multiplication, one obtains

(2.138)

or else

(2.139)

after lumping powers of x and resorting to a more condensed notation. Of particular interest is having n = m = 1, besides b 0 = − a 0and a 1 = b 1 = 1 – in which case Eq. (2.139)takes the form

(2.140)

Eq. (2.140)entails a notable case of multiplication – since the product of two conjugated binomials, i.e. x + a 0as per Eq. (2.135)and x − a 0as per Eq. (2.136), equals the difference of the squares of their bases, i.e. x 2− a 0 2. This mathematical feature is useful when the terms under scrutiny are square roots (since the product of two irrational functions would turn to a rational function).

With regard to division of a dividend polynomial, say, P n{ x }, by a divisor polynomial, say, P m{ x }, one will eventually be led to

(2.141)

on account again of Eqs. (2.135)and (2.136)– provided that n ≥ m ; here Q n−mdenotes an ( n – m )th degree quotient polynomial, and R denotes a remainder polynomial of degree not exceeding m . The underlying algorithm is but an extension of Euclidean (long) division algorithm for regular numbers; one should thus start by dividing the highest order term, a n x n, of the dividend polynomial by the highest order term, b m x m, of the divisor polynomial – so a n x n−m/ b malways appears as first term of the quotient polynomial; multiplication should then proceed of said quotient term by every term of the divisor – followed by subtraction of the result from the dividend. In other words, the first step of division should lead to

(2.142)

which simplifies to

(2.143)

after lumping factors and canceling a n x nwith its negative afterward; this is graphically illustrated in Fig. 2.7.The same algorithm may now be applied to the second term in the right‐hand side of Eq. (2.143), as long as n – 1 ≥ m , to produce