:
Tom Kelliher, MA 114
Nov. 16, 1998
Collect homework. Projects due Friday.
Will be out week after Thanksgiving.
Read 4.4
Homework due Monday 11/30. Section 4.2: 24, 26, 28, 38, 41, 43. Trig. extra credit. 6.4: 12, 40, 52, 56, 64, 70.
A method for finding the roots of a polynomial :
. Graph 
to see which numbers on the list could possibly be
roots and test each one to find the rational roots.
as a product of
linear factors (one for each rational root) and another factor 
.
are the roots of 
. If 
is
quadratic, use the quadratic formula.
has degree 3 or more, check for repeated rational roots by
evaluating 
at each of the rational roots found in step 1.
as a product of linear factors and a
factor 
which has no rational roots.
for 
.
are the roots of 
. Use the
bounds test to find upper and lower bounds for its roots and then
graphically approximate these roots.
What is a rational number? Examples of irrationals: 
,
.
If a rational number(in lowest terms) is a root of the polynomial
with integer coefficients and
and
then:
r is a factor of
and
s is a factor of
.
Example: What are the possible rational roots of 
? Which ones are roots? Graph, visually eliminate, test.
Once you have a root, what is the corresponding linear factor?
as a product of linear factors and 
.
quadratic? If so, then apply quadratic formula.
. If so, simplify 
and
keep checking.
Consider 
. Graphing, we hypothesize the roots
lie between -1 and 3.
Use synthetic division and a sign argument to prove.
Letbe a polynomial with positive leading coefficients.
If d > 0 and every number in the last row in the synthetic division of
by x - d is nonnegative, then d is an upper bound for the real roots of
.
If c < 0 and the numbers in the last row of the synthetic division of
by x - c are alternately positive and negative (with 0 considered as either), then c is a lower bound for the real roots of
.
Pg. 228: 23, 25, 27...
