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Zeros of Polynomial Functions
Section 2.5
Page 312
Review


Factor Theorem: If x – c is a factor of
f(x), then f(c) = 0
Example: x – 3 is a zero of
f(x) = 2x3 – 3x2 – 11x + 6
Information about
The Rational Zero Theorem
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

Use to find possible rational zeros
Provides a list of possible rational zeros
of a polynomial function
Not every number will be a zero
The Rational Zero Theorem

If f(x) has integer coefficients and p/q
(where p/q is reduced to lowest terms)
is a rational zero of f, the p is a factor
of the constant term a0 and q is a factor
of the leading coefficient an
Possible zeros = factors of a0 = p
factors of an q
Example 1

List all possible zeros of f(x) = -x4 + 3x2 + 4
Example 2

List all possible zeros of
f(x) = 15x3 + 14x2 – 3x – 2
Finding zeros

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Use the Rational Zero Theorem and trial
& error to find a rational zero
Once the polynomial is reduced to a
quadratic then use factoring or the
quadratic formula to find the remaining
zeros.
Example 3

Find the zeros of f(x) = x3 + 2x2 – 5x – 6
Example 4

Find all the zeros of
f(x) = x3 + 7x2 + 11x – 3
Example 5

Solve: x4 – 6x2 – 8x + 24 = 0
Practice

List all possible rational zeros
1.

Find all zeros
2.
3.
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f(x) = 4x5 + 12x4 – x – 3
f(x) = x3 + 8x2 + 11x – 20
f(x) = x3 + x2 – 5x – 2
Solve
4.
x4 – 6x3 + 22x2 – 30x + 13 = 0
Properties of Polynomial
Equations
If a polynomial equation is of degree, n, then
counting multiple roots separately, the equation
has n roots.
If a + bi is a root of a polynomial equation with
real coefficients (b ≠ 0), then the complex
imaginary number a – bi is also a root. Complex
imaginary roots, if they exist, occur in conjugate
pairs.
1.
2.
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
If 3i is a root, then –3i is also a root
If 2 – 5i is a root, then 2 + 5i is also a root
The Fundamental Theorem of
Algebra

If f(x) is a polynomial of degree n, where n ≥ 1,
then the equation f(x) = 0 has at least one
complex root.
Example 6

Find a 4th degree polynomial function
f(x) with real coefficients that has -2, 2,
and i as zeros and such that f(3) = -150
Descartes’s Rule of Signs
1.
The number of positive real zeros of f is either
a.
b.
The same as the number of sign changes of f(x) OR
Less than the number of sign changes by an even integer
Note: if f(x) has only one sign change, then f has only
one positive real zero
2. The number of negative real zeros of f is either
a.
b.
The same as the number of sign changes of f(-x) OR
Less than the number of sign changes by an even integer
Note: if f(-x) has only one sign change, then f has only
one negative real zero
Review
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Table on page 320
Negative Real Zeros
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f(-x) = -3x7 + 2x5 – x4 + 7x2 – x – 3
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f(-x) = -4x5 + 2x4 – 3x2 – x + 5
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f(-x) = -7x6 - 5x4 – x + 9
Example 7

Determine the possible numbers of
positive and negative real zeros of
f(x) = x3 + 2x2 + 5x + 4
Practice


Find a third-degree polynomial function
with real coefficients that has -3 and i
as zeros such that f (1) = 8
Determine the possible numbers of
positive and negative real zeros of f(x)
= x4 - 14x3 + 71x2 – 154x + 120