after lumping factors and canceling an xn with its negative afterward; this is graphically illustrated in Fig. 2.7.The same algorithm may now be applied to the second term in the right‐hand side of Eq. (2.143), as long as n – 1 ≥ m, to produce
(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 Pn would be an exact (polynomial) multiple of Pm .
Figure 2.7 Graphical algorithm of (long) Euclidean division of polynomials on x – where …, an−2, an−1, an , …, bm−2, bm−1, and bm denote real numbers, while n and m denote integer numbers.
In the particular case Pm is a linear polynomial, of the form x − r, viz.
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,
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, …, an−2, an−1, an, 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 Pn {x}|x = r, see Eq. (2.135). Said remainder will be zero, i.e.
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 Pn {x} exactly – or Pn {x} is a multiple of x − r, when x = r is a root of Pn {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),
Eq. (2.159) indicates
