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The synthetic form of the division is as follows:
The steps in this synthetic form of the division are explained in the treatment
of the general case which follows.
The synthetic division of
p(x) = a0xn +
a1xn -1 + ... + an
-1x + an
by x - h is in the form
where c0 = a0, b1
= hc0, c1 = a1 +
b1, b2 = hc1, c2
= a2 + b2, ..., bn
= hcn -1, cn =
an + bn. In general, each b
is h times the previous c, c0 = a0,
and each succeeding c is the sum of the a and b above
it. The last c, cn , is the value of
p(h), and the other c's are the coefficients of q(x)
in the formula p(x) = (x - h)q(x)
+ p(h); they give us the expression
p(x) = (x - h)(c0xn
-1 + c1xn -2 + ...
+ cn -2x + cn
-1) + cn .
Example 1. Express p(x) = x5
+ 25x2 + 7 in the form (x + 3)q(x)
+ p(-3).
Solution: We note that h = -3 and that a0
= 1, a1 = 0, a2 = 0, a3
= 25, a4 = 0, and a5 = 7 in this problem.
The synthetic division is therefore written
Hence:
x5 + 25x2 + 7 = (x +
3)(x4 - 3x3 + 9x2
- 2x + 6) - 11.
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Tuesday, June 2, 1998