Download Section 3.4 Zeros of Polynomial Functions *Rational Zero Theorem If

Section 3.4 Zeros of Polynomial Functions
*Rational Zero Theorem
p
p
(where
is reduced
q
q
to lowest terms) is rational zero of f , then p is a factor of the constant term, a 0 , q is a
factor of the leading coefficient, a n .
Factors of a0
Possible rational zeros=
.
Factors of an
n
n 1
If f ( x)  a n x  a n 1 x    a1 x  a 0 has integer coefficient and
3
2
Example 1) List all possible rational zeros of f ( x)  x  2 x  5x  6 .
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Example 2) List all possible rational zeros of f ( x)  4 x  12 x  x  3 .
*Finding Zeros of a Polynomial Function
Step 1 List all possible rational zeros
Step 2 Use synthetic division to find a rational zero among the possibilities.
Step 3 Factor the polynomial using synthetic division.
Step 4 Repeat Step 1 through Step 3 until you get a quadratic factor.
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Example 3) Find all zeros of f ( x)  x  8 x  11x  20 .
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3
2
Example 4) Find all zeros of f ( x)  x  x  5x  2 .
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3
2
Example 5) Solve x  6 x  22 x  30 x  13  0 .
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*Properties of Polynomial Equations
1. If a polynomial equation is of degree n, then counting multiple roots separately, the
equation has n roots.
2. If a  bi is a root of a polynomial equation with real coefficient (b  0) , then the
complex imaginary number a  bi is also a root. Complex imaginary roots, if they exist,
occur in conjugate pairs.
*The Linear Factorization Theorem
n
n 1
If f ( x)  a n x  a n 1 x    a1 x  a 0 , where n  1 and an  0 , then
f ( x)  an ( x  c1 )( x  c2 )( x  cn ) ,
Where c1 , c2 ,, cn are complex numbers (possibly real and not necessarily distinct).
Example 6) Find a third-degree polynomial function f (x) with real coefficients that has -3
and i as zeros and such that f (1)  8 .
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