58 2. Evaluation, Division, and Expansion
If we think of the “size” of a polynomial as being measured by its de-
gree, then, as with numbers, we can take away a multiple of g(t) which
leaves as a remainder, a “smaller” quantity which will in turn be sub-
sequently divided by g(t). In the context of polynomials, “smaller”
means “of lower degree”. Subtracting 4t2g(t) will leave such a remain-
der.
(4
(b)
cc>
Verify that f(t) - 4t2g(t) = -31t4 - 12t3 + t2 + 6.
Take away a multiple of g(t) from the right hand side of (a) to
leave a remainder of lower degree. Continue on in this fashion
until a remainder of degree less than that of g(t) is obtained.
Show that the process of parts (a) and (b) can be written in the
form of the long division algorithm:
4tz - 31t + 205
t3 + 7tz + 3t - 21 4t5 - 3t4 + OP - 7P + Ot+ 6
’ 4t5 + 28t4 -i- 12t3 - 8t2
- 31t4 - 12t3 + t2 + Ot+ 6
- 31t4 - 217t3 - 93t2 + 62t
205t3 -I- 94t2 - 62t + 6
205t3 + 1435t2 + 615t - 410- 1341t2 - 677t + 416
(4
(e)
From the algorithm in (c), read off the quotient and the remain-
der for the equation
f(t) = g(W) + r(t).
Check your answer by directly computing the right hand side.
The algorithm in (c) can be more clearly presented by suppress-
ing the variable to obtain
4 -31 205
173 -2)4 -3 0 -7 0 6
4 28 12 -8
-31 -12 1
-31 -217 -93 62
205 94 -62
205 1435 615 -410
-1341 -677 416
A further compression of this is Homer’s Method of Synthetic Di-
vision which can be regarded as a generalization of his method for
division by a binomial (t - c) and which in this case takes the form: