Download f(x) = 2x4+7x3-4x2-27x-18 a. Is (x-5) a likely factor of f

January 13, 2012
Warm Ups 16-5 Precalc
f(x) = 2x4+7x3-4x2-27x-18
a.
b.
c.
d.
Is (x-5) a likely factor of f(x)?
Is (3x+2) a likely factor of f(x)?
Is (x-2) a likely factor of f(x)?
factor f(x) completely
January 13, 2012
DESCARTES'S RULE OF SIGNS:
1. the number of POSITIVE REAL ZEROs is either equal to
the number of variations in sign of f(x) or less than that by an
even integer.
2. the number of NEGATIVE REAL ZEROS is either equal to
the number of variation in signs of f(-x) or less than that by
an even integer.
f(x) = 2x4+7x3-4x2-27x-18 had four zeros: 2,-3, -1, and -3/2
f(x) has 1 change in sign, which indicates that it has 1 positive
real zero.
f(-x) = 2(-x)4 + 7(-x)3 -4(-x)2-27(-x) - 18
= 2(x)4 - 7( x)3 -4(x)2 +27x - 18
therefore f(-x) has 3 changes in sign, indicating that it has 3
or 1 negative real zeros.
January 13, 2012
HOW DOES THIS HELP US SOLVE HIGHER DEGREE
EQUATIONS??
Use Descartes's rule of signs to find all solutions to
x4-3x3+x2+3x-2 = 0
January 13, 2012
We have also been also taking advantage of the rational
zero test
Rational Zero Test:
If f(x) = anxn + an-1xn-1 + ...+a1x + a0 has integer
coefficients, then every rational zero of f is p/q, where
p is a factor of constant term a0 and q is a factor of the
leading coefficient an
That looks more complicated than it is! It is just
observing that if f(x) = 2x3+ 4x -5, then (3x+7) isn't a
good factor guess! Which means -7/3 isn't going to be a
zero.
January 13, 2012
List all of the possible rational zeros of f. Using those
and Descartes's rule of signs, find all of zeros and graph
f(x)= -3x3+20x2-36x+16
January 13, 2012
One last trick to have up your sleeves......
LOWER AND UPPER BOUND RULE
let f(x) be a polynomial with real coefficients and a positive
leading coefficient. Suppose f(x) is divided by (x-c) using
synthetic division, and it wasn't a factor.
1. If c>0 and each number in the last row is either
positive or zero, then c is an UPPER BOUND for the zeros
2. If c<0 and the numbers in the last row are alternately
positive and negative (0 counts either way), then c is a
LOWER BOUND for the zeros
January 13, 2012
Let's use this on finding the real zeros of
f(x) = 3x3-2x2+15x-10
3 variations of signs, so either 3 or 1 positive real zeros
f(-x) = -3x3-2x2-15x-10
no variations of signs, so no negative real zeros
Try 1. If it is an upperbound, what does this tell us?
January 13, 2012
#16-5 p. 211 box, 53
January 13, 2012
January 13, 2012
January 13, 2012
January 13, 2012