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

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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 denote 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 = r₁ is a root of P_n {x}, then

      (2.166)

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

      (2.167)

      – so x − r₁ may be factored out to produce

(2.168)

therefore, r₁ being a root of P_n implies indeed that x − r₁ is 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 − r₁, 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 nth‐ 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 b₁, b₂, …, b_n−2 denote 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 r₂ of P_n−1{x}, abiding to

      (2.176)

then Eq. (2.175) simplifies to

(2.177)

since x − r₂ appears in all terms of Eq. (2.177), it may be factored out to yield

(2.178)

– so insertion of Eq. (2.178) transforms

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