Читать книгу Mathematics for Enzyme Reaction Kinetics and Reactor Performance - F. Xavier Malcata - Страница 23

Recalling the two polynomials labeled as Eqs. (2.135) and (2.136) – of nth and mth degree, respectively, one may define their product P_n P_m as 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 b₀ = −a₀ and a₁ = b₁ = 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 + 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).