Download Zeros of Polynomial Functions

Zeros of Polynomial
Functions
Section 2.5
Objectives
• Use the Factor Theorem to show that x-c is a
factor a polynomial.
• Find all real zeros of a polynomial given one or
more zeros.
• Find all the rational zeros of a polynomial
using the Rational Zero Test.
• Find all real zeros of a polynomial using the
Rational Zero Test.
• Find all zeros of a polynomial.
• Write the equation of a polynomial given some
of its zeros.
Vocabulary
• rational zero
• real zero
• multiplicity
Factor Theorem
Let f (x) be a polynomial
a. If f(c) = 0, then x – c is a factor of f (x).
b. If x – c, is a factor of f(x), then f(c) = 0.
If c = 3 is a zero of the
polynomial
P (x )  x  17x  90x  144,
3
2
find all other zeros of P(x).
Use synthetic division to show
that x = 6 is a solutions of the
equation
x  5x  56x  60  0
3
2
Rational Root (Zero) Theorem
(Test)
If
f (x )  an x n  an 1x n 1    a1x  a0
p
p
has integer coefficients and
(where
is
q
q
reduced to lowest terms) is a rational zero of f,
then p is a factor of the constant term, a0, and
q is a factor of the leading coefficient, an.
Find all the rational zeros of the
polynomial
P (x )  x  3x  x  3
3
2
Find all the real zeros of the
polynomial
P (x )  x  5x  2
3
Linear Factorization Theorem
If f (x )  an x n  an 1x n 1    a1x  a0 ,
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).
Factor
3
2
P (x )  x  x  3x  9
into linear and irreducible
quadratic factors with real
coefficients.
Find all the zeros of the
polynomial
P (x )  x  x  3x  9
3
2
Find all the zeros of the
polynomial
P (x )  x  2x  x
5
3
Find the equation of a
polynomial of degree 4 with
integer coefficients and
leading coefficient 1 that had
zeros x = -2-3i, and at x = 1
with x = 1 a zero of
multiplicity 2.
Descartes’s Rule of Signs
Let f (x )  an x n  an 1x n 1    a1x , a0
Be a polynomial with real coefficients.
1. The number of positive real zeros of f is either
a.
the same as the number of sign changes of f(x)
OR
b.
less than the number of sign changes of f(x) by a
positive even integer.
If f(x) has only one variation in sign, then f has exactly
one positive real zero.
Descartes’s Rule of Signs
Let f (x )  an x n  an 1x n 1    a1x , a0
Be a polynomial with real coefficients.
1. The number of negative real zeros of f is either
a.
the same as the number of sign changes of f(—x)
OR
b.
a
less than the number of sign changes of f(—x) by
positive even integer.
If f(—x) has only one variation in sign, then f has
exactly one negative real zero.
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.