Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier Malcata
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(2.129)
where minus signs may be dropped from numerator and denominator,
(2.130)
upon division of both numerator and denominator of the last term by 1 − k2, 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 k2 = 1 in Fig. 2.6 with the lines in Fig. 2.4 corresponding to the same
(2.133)
based on Eq. (2.110) and matching Eq. (2.76).
2.2 Multiplication and Division of Polynomials
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
and
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.
2.2.1 Product
Recalling the two polynomials labeled as Eqs. (2.135) and (2.136) – of nth and mth degree, respectively, one may define their product Pn Pm as an (n + m)th degree polynomial, viz.
(2.137)
upon application of the distributive property of multiplication, one obtains
(2.138)
or else
after lumping powers of x and resorting to a more condensed notation. Of particular interest is having n = m = 1, besides b0 = −a0 and a1 = b1 = 1 – in which case Eq. (2.139) takes the form
Eq. (2.140) entails a notable case of multiplication – since the product of two conjugated binomials, i.e. x + a0 as per Eq. (2.135) and x − a0 as per Eq. (2.136), equals the difference of the squares of their bases, i.e. x2 − a02. 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).
2.2.2 Quotient
With regard to division of a dividend polynomial, say, Pn {x}, by a divisor polynomial, say, Pm {x}, one will eventually be led to
on account again of Eqs. (2.135) and (2.136) – provided that n ≥ m; here Qn−m denotes an (n – m)th degree quotient polynomial, and R<m 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, an xn, of the dividend polynomial by the highest order term, bm xm, of the divisor polynomial – so an xn−m /bm always 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