Download 7.5 Descartes` Rule of Signs

2.5 Descartes’ Rule of Signs
• To apply theorems about the zeros of
polynomial functions
• To approximate zeros of polynomial
functions
Descartes’ Rule of Signs
• The number of positive real zeros of a
polynomial function P(x), with real
coefficients, is equal to the number of
variations in sign of the terms of P(x) or
is less than this number by a multiple of
2.
• The number of negative real zeros is
equal to the number of variations in sign
of the terms of P(-x) or is less than this
number by a multiple of 2.
Example 1
• Find the possible number of positive
and negative real zeros of
f ( x)  3x  5x  2 x  x  10
4
3
2
The problem isn’t asking for the zeros
themselves, but what are the possible
number of them. This can help narrow
down the possibilities when you do go on
to find the zeros.
Example 2
• Find the possible number of positive and
negative real zeros of
f ( x)  2 x  7 x  8
3
Upper and Lower Bounds
Let P(x), a polynomial function with positive
leading coefficient, be divided by x – c.
If c > 0 and all coefficients in the quotient
and remainder are nonnegative, then c is
an upper bound of the zeros.
If c < 0 and the coefficients in the quotient
and remainder alternate in sign, then c is a
lower bound of the zeros.
Example 3
• Show that all real roots of the equation
x  5 x  10 x  12 x  20  0
5
3
2
Lie between –4 and 4
The Intermediate Value
Theorem
• If f(x) is a polynomial function and f(a)
and f(b) have different signs, then
there is at least one value, c, between
a and b such that f(c)=0.
• This works because 0
separates the positives from the
negatives. So to go from
positive to negative or vice a
versa you would have to hit a
point in between that goes
through 0.
Example 4
Show that
f ( x)  x  3 x  6 x  2
4
3
has real zeros between 3 and 4 and
0 and -1.
Example 5
Find the upper and lower bounds.
Find the possible number of positive and
negative zeros.
Find the location of the zeros.
f ( x)  x  3 x  3 x  4
3
2
Example 6
Find the upper and lower bounds.
Find the possible number of
positive and negative zeros.
Find the location of the zeros.
f ( x)  x  3x  x  12 x  20
4
3
2
Summary
• Descartes’ Rule of Signs is used to
determine the number of possible positive
and negative zeros.
• The Upper and Lower Bound Theorem is
used to determine the bounds of the
zeros.
• The Intermediate Value Theorem is used
to determine between which real numbers
a zero occurs.