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

(2.144)

that degenerates to

(2.145)

after lumping the coefficients of similar powers of x – and the process may be iterated for the third term, the fourth term, and so on. The above algorithm is thus to be repeated until the degree of the polynomial in numerator of the last term is lower than its denominator counterpart; it may even reduce to zero – in which case P nwould be an exact (polynomial) multiple of P m.

Figure 2.7 Graphical algorithm of (long) Euclidean division of polynomials on x – where …, a n−2, a n−1, a n, …, b m−2, b m−1, and b mdenote real numbers, while n and m denote integer numbers.

In the particular case P mis a linear polynomial, of the form x − r , viz.

(2.146)

en lieu of Eq. (2.136)and meaning that b 0 = −r and b 1 = 1, the algorithm of division of polynomials simplifies to

(2.147)

which breaks down to

(2.148)

upon condensation of terms alike; a second application of said algorithm unfolds

(2.149)

where a similar condensation of powers alike gives rise to

(2.150)

This process may then be iterated until the numerator of the last term reduces to a constant – according to

(2.151)

– or, after having eliminated inner parentheses,

(2.152)

a graphical illustration – often known as Ruffini’s rule, is depicted in Fig. 2.8. Except for the first column – where the lower entry mimics the upper entry, the lower entry of every column entails the product of the previous counterpart by r , added to the current upper entry.

Figure 2.8 Graphical algorithm of (long) Ruffini’s division of polynomials – where a 0, a 1, a 2, …, a n−2, a n−1, a n, and r denote real numbers, while n denotes an integer number.

A more condensed notation is, however, possible based on Eq. (2.152), viz.

(2.153)

– by taking advantage of the concept of summation; note that the numerator of the last term coincides with P n{ x }| x = r, see Eq. (2.135). Said remainder will be zero, i.e.

(2.154)

when

(2.155)

Eq. (2.155) may be rephrased as

(2.156)

or, in view of Eq. (2.135),

(2.157)

Therefore, the linear polynomial x − r divides P n{ x } exactly – or P n{ x } is a multiple of x − r , when x = r is a root of P n{ x }.

An alternative statement of the result conveyed by Eq. (2.154)is

(2.158)

at the expense of Eq. (2.141)– or, in view of Eqs. (2.135)and (2.146),

(2.159)

Eq. (2.159) indicates that x − r will be a factor of polynomial P nwhenever r is itself a root of P n. This very same conclusion may be achieved after recalling that a function, f { x }, may in general be represented by an infinite series on x , i.e.

(2.160)

according to Taylor’s theorem (to be derived in due course) – where ξ denotes any point of the interval of definition of f { x }; in the particular case of an n th‐degree polynomial, the said expansion becomes finite and entails only n + 1 terms, according to

(2.161)

with the aid of Eq. (2.135), for the simple reason that d n+1 P n/ dx n+1 = d n+2 P n/ dx n+2 = ⋯ = 0. Under such circumstances, Taylor’s coefficients look like

(2.162)