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

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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_m denote real numbers, while n and m denote integer numbers.

      In the particular case P_m is a linear polynomial, of the form x − r, viz.

(2.146)

      en lieu of Eq. (2.136) and meaning that b₀ = −r and b₁ = 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 a₀, a₁, a₂, …, 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}.

      2.2.3 Factorization

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_n whenever 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 nth‐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 dⁿ⁺¹ P_n /dxⁿ⁺¹ = dⁿ⁺² P_n /dxⁿ⁺² = ⋯ = 0. Under such circumstances, Taylor’s coefficients look like

      (2.162)

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

(2.163)

      where a₁, a₂, …, a_n−1