Zeros of Polynomial Functions

1
Zeros of Polynomial Functions

• Section 2.5

2
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.

3
Vocabulary

• rational zero
• real zero
• multiplicity

4
Factor Theorem

• Let f (x) be a polynomial
• If f(c) 0, then x c is a factor of f (x).
• If x c, is a factor of f(x), then f(c) 0.

5
If c 3 is a zero of the polynomial find all
other zeros of P(x).
6
Use synthetic division to show that x 6 is a
solutions of the equation
7
Rational Root (Zero) Theorem (Test)
8
Find all the rational zeros of the polynomial
9
Find all the real zeros of the polynomial
10
Linear Factorization Theorem
If
, where n 1 and an ? 0, then Where
c1, c2, . . ., cn are complex numbers (possibly
real and not necessarily distinct).
11
Factor into linear and irreducible quadratic
factors with real coefficients.
12
Find all the zeros of the polynomial
13
Find all the zeros of the polynomial
14
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.
15
Descartess Rule of Signs

• Let
  ,
• Be a polynomial with real coefficients.
• 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.
16
Descartess Rule of Signs

• Let
  ,
• Be a polynomial with real coefficients.
• The number of negative 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 negative real zero.
17
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.