Download Example 1: Determine the possible number of positive and negative

7.5 Descartes' Rule, the Intermediate Value Theorem, and the Sum & Product
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: Determine the possible number of positive and negative real zeros of the
polynomial P(x) = x3 - 5x2 + 4x + 12.
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Location Theorem:
If P(x) is a polynomial function with real coefficients, and a and b are real numbers
such that P(a) is positive and P(b) is negative, then P(x) has at least one zero
between a and b.
Intermediate Value Theorem: If P(x) is a polynomial function such that a < b and
P(a) ≠ P(b), then P(x) takes on every value between P(a) and P(b) in the interval [a,b].
Example 2: Graph f(x) = x3 + x - 1 using a graphing calculator and locate the real zero
to the nearest hundredth.
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Sum and Product of Zeros Theroem:
In the polynomial f(x) = a nxn + a n-1 xn-1 + a
of the zeros is equal to -a n-1 .
an
The product of the zeros is equal to
n-2
xn-2 + ... + a
1
x+a
0,
a 0 ≠ 0, the sum
a 0 for n even and -a 0 for n odd.
an
an
Example 3: Determine the sum and product of the zeros for each of the following:
a. f(x) = 4x4 - 3x3 + 2x2 - x + 6
b. g(x) = 3x5 - 2x3 + x2 - x + 4
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Upper Bound - a number greater than or equal to the greatest zero of the function.
Lower Bound - a number less than or equal to the least real zero of the function.
Upper and Lower Bound Theorem:
Let P(x), a polynomial function with positive leading coefficient, be divided by x-c.
- If c > 0 and all the 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 4: Determine the least integral upper bound and the greates integral lower
bound for f(x) = x5 + 5x4 - 3x3 - 29x2 + 2x + 23.
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Example 5:
Sketch a graph of P(x) that satisfies the following conditions:
• P(x) is 4th degree with leading coefficient 1
• P(x) has two distinct negative real zeros and one positive real zero with multiplicity
2.
• The greatest integral lower bound for the zeros of P(x) is -5.
• The least integral upper bound is 4.
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