Download Pre-Calculus I 2.5 Zeros of Polynomial Function Recall: Natural or

Pre-Calculus I
2.5 Zeros of Polynomial Function
Recall:
{1,2,3,4,…}
{0,1,2,3,4,…}
{…,-3,-2,-1,0,1,2,3,…}
Natural or Counting Numbers (N)
Whole Numbers (W)
Integers (Z)
Rational Numbers (Q)
Real Numbers (R)
Irrational Numbers
Examples
Z
W
N
R
Q
The Rational Zero Theorem –
If
has integer coefficients
and (where is reduced to lowest terms) is a rational zero of f,
then p is a factor of the constant term
leading coefficient, .
and q is a factor of the
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Example – List all possible rational zeros of
Example - List all possible rational zeros of
and find the zeros of f(x).
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Example – Solve:
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Properties or Roots of Polynomial Equations
1. If a polynomial equation is of degree n, then counting
multiple roots separately, the equation has n roots.
2. If
is a root of a polynomial equation with real
coefficients (
), then the imaginary number
is also
a root. Imaginary roots, if they exist occur in conjugate pairs.
For example if
is a zero of a polynomial function f(x) with
real coefficients, then the conjugate
is also a zero.
The Fundamental Theorem of Algebra –
If
is a polynomial of degree , where
, then the
equation
has at least one complex root (that is at least
one zero in the system of complex numbers). Note: that complex
numbers include both real numbers
and imaginary numbers
The Linear Factorization Theorem –
If
, then
are complex numbers.
where
and
where
In words: An nth degree polynomial can be expressed as the
product of a nonzero constant and n linear factors, where each
linear factor has a leading coefficient of 1
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Example – Find a polynomial function of degree 3 having zeros
1,3i and -3i
____
____
____
Example – Find a polynomial function of degree 5 with -1 as a
zero of multiplicity 3, 4 as a zero of multiplicity 1, and 0 a zero of
multiplicity 1
____
____
____
____
____
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Example – Find a fourth degree polynomial function with real
coefficients that has -2, 2 and as zeros and such that
.
____
____
____
____
Descartes’s Rule of Signs –
Let P(x), written in decreasing exponential order or ascending
exponential order be a polynomial function with real coefficients
and a nonzero constant term.
The number of positive real zeros of P(x) is either:
1. the same as the number of variations of sign in P(x)
2. Less than the number of sign variations of sign in P(x),
by a positive even integer. If P(x) has only one
variation in sign then P has exactly one positive real
zero.
The number of negative real zeros of P(x) is either:
1. the same as the number of variations of sign in P(-x)
2. Less than the number of sign variations of sign in P(-x),
by a positive even integer. If P(x) has only one
variation in sign then P has exactly one negaitve real
zero.
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Example – Determine the possible number of positive and
negative real zeros of
Step 1:
How many sign changes? ____
_____ Positive Real Zeros
Step 2:
How many sign changes? ____
_____ Negative Real Zeros
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Hannah Province – Mathematics Department Southwest Tn Community College
Example – Determine the possible number of positive and
negative real zeros of
Step 1:
How many sign changes? ____
_____ Positive Real Zeros
Step 2:
How many sign changes? ____
_____ Negative Real Zeros
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Hannah Province – Mathematics Department Southwest Tn Community College