Professor P. Bishop
MAC1147
(MDN)
R.6 Synthetic Division (Condensed)
Synthetic division
is used to divide a polynomial of degree 1 or higher by a binomial
g(x
)
of the form
x – c
.
Use synthetic division to determine whether x – c is a factor of f(x):
1.
(
29
(
29
=
+

+

=
+
f
x
x
x
x
x
; g
x
x
4
3
2
8
1 5
4
4
2.
(
29
(
29
=
+

=

f
x
x
x
; g
x
x
2
3
5
4
5.5 The Real Zeros of a Polynomial Function
Remainder Theorem:
If f(x) is divided by xc, then the remainder is f(c).
Factor theorem:
If
f
is a polynomial
function, then
x – c
is a factor of
f(x)
if any only if
f(c) = 0
.
Number of Real Zeros Theorem:
A polynomial function
cannot have more real zeros than its degree.
Descartes’ Rule of Signs:
If
f
is a polynomial function in standard form,
then (a) the number of positive real zeros of
f
either equals the number of variations in the sign of the nonzero
coefficients of
f(x)
or else equals that number less an even integer;
(b) ) the number of negative real zeros of
f
either
equals the number of variations in the sign of the nonzero coefficients of
f(x)
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 Fall '08
 Staff
 Division, Synthetic Division, Binomial, Continuous function, Complex number, Professor P. Bishop

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