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2.4
Real Zeros of Polynomial Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
Rewrite the expression as a polynomial in standard form.
2 x  3x  x
1.
x
2 x  8x  x
2.
2x
Factor the polynomial into linear factors.
3
2
5
3
2
2
3. x  16 x
3
4. x  x  4 x  4
5. 6 x  24
3
2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 2
Quick Review Solutions
Rewrite the expression as a polynomial in standard form.
2 x  3x  x
1.
2 x  3x  1
x
2 x  8x  x
1
2.
x  4x 
2x
2
Factor the polynomial into linear factors.
3
2
2
5
3
2
3
2
3. x  16 x
3
x  x  4  x  4 
 x  1 x  2  x  2 
6  x  2  x  2 
4. x  x  4 x  4
3
2
5. 6 x  24
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 3
What you’ll learn about





Long Division and the Division Algorithm
Remainder and Factor Theorems
Synthetic Division
Rational Zeros Theorem
Upper and Lower Bounds
… and why
These topics help identify and locate the real zeros of
polynomial functions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 4
Division Algorithm for Polynomials
Let f ( x) and d ( x) be polynomials with the degree of f
greater than or equal to the degree of d , and d ( x)  0.
Then there are unique polynomials q( x) and r ( x), called
the quotient and remainder, such that
f ( x)  d ( x)  q ( x)  r ( x) where either r ( x)  0 or the
degree of r is less than the degree of d .
The function f ( x) in the division algorithm is the dividend,
and d ( x) is the divisor. If r ( x)  0, we say d ( x) divides
evenly into f ( x).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 5
Division Algorithm for Polynomials
Symbolically, we would represent this as
q( x)  r ( x)
f ( x)
 d ( x)
f ( x)
d ( x)
where either r ( x)  0 or the degree of r is less than the
degree of d .
4+R3
Numerically, we would write, e.g., 4
19
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 6
Example Using Polynomial Long Division
Use long division to find the quotient and remainder
when 2 x 4  x3  3 is divided by x 2  x  1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 7
Remainder Theorem
If polynomial f ( x) is divided by x  k , then the remainder is r  f (k ).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 8
Example Using the Remainder Theorem
Find the remainder when f ( x)  2 x 2  x  12 is divided by x  3.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 9
Factor Theorem
A polynomial function f ( x) has a factor x  k if and
only if f (k )  0.
In this case, k is a real zero or a real root of f  x  .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 10
Example Using Synthetic Division
Divide 3 x3  2 x 2  x  5 by x  1 using synthetic division.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 11
Descartes’ Rule of Signs
Let P(x) be a polynomial over the real numbers.
The number of positive roots of P(x) = 0 either equals
the number of sign variations in P(x) or else is fewer by
an even number.
The number of negative roots of P(x) = 0 either equals
the number of sign variations in P(–x) or else is fewer
by an even number.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 12
Rational Zeros Theorem
Suppose f is a polynomial function of degree n  1
of the form f ( x)  an x n  an1 x n1  ...  a0 , with
every coefficient an integer and a0  0. If x  p / q
is a rational zero of f , where p and q have no
common integer factors other than 1, then
p is an integer factor of the constant coefficient a0 , and
q is an integer factor of the leading coefficient an .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 13
Upper and Lower Bound Tests for Real
Zeros
Let f be a polynomial function of degree n  1 with a
positive leading coefficient. Suppose f ( x) is divided
by x  k using synthetic division.
If k  0 and every number in the last line is nonnegative
(positive or zero), then k is an upper bound for the real
zeros of f .
If k  0 and the numbers in the last line are alternately
nonnegative and nonpositive, then k is a lower bound
for the real zeros of f .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 14
Example Finding the Real Zeros of a
Polynomial Function
Find all of the real zeros of f ( x)  2 x 4  7 x3  8 x 2  14 x  8.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 15
Example Finding the Real Zeros of a
Polynomial Function
Find all of the real zeros of f ( x)  3x 4  2 x 3  3x 2  x  2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 16
Homework



Homework #1
Read Section 2.5
Page 223, Exercises: 1 – 73 (EOO), skip 69
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 17
2.5
Complex Zeros and the Fundamental
Theorem of Algebra
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
What you’ll learn about



Two Major Theorems
Complex Conjugate Zeros
Factoring with Real Number Coefficients
… and why
These topics provide the complete story about
the zeros and factors of polynomials with real number
coefficients.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 19
Fundamental Theorem of Algebra
A polynomial function of degree n has n
complex zeros (real and imaginary). Some of
these zeros may be repeated.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 20
Linear Factorization Theorem
If f ( x) is a polynomial function of degree n  0, then
f ( x) has precisely n linear factors and
f ( x)  a ( x  z )( x  z )...( x  z )
1
2
n
where a is the leading coefficient of f ( x) and
z , z ,..., z are the complex zeros of f ( x).
1
2
n
The z are not necessarily distinct numbers; some may
i
be repeated.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 21
Fundamental Polynomial Connections in
the Complex Case
The following statements about a polynomial
function f are equivalent, if k is a complex
number:
1. x = k is a solution (or root) of the equation f(x) = 0
2. k is a zero of the function f.
3. x – k is a factor of f(x).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 22
Example Exploring Fundamental
Polynomial Connections
Write the polynomial function in standard form, identify
the zeros of the function and the x-intercepts of its graph.
f ( x)  ( x  3i )( x  3i )
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 23
Complex Conjugate Zeros
Suppose that f ( x) is a polynomial function with real
coefficients. If a and b are real numbers with b  0
and a  bi is a zero of f ( x), then its complex conjugate
a  bi is also a zero of f ( x).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 24
Example Finding a Polynomial from
Given Zeros
Write a polynomial of minimum degree in standard form
with real coefficients whose zeros include 2,  3, and 1  i.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 25
Factors of a Polynomial with Real
Coefficients
Every polynomial function with real
coefficients can be written as a product of
linear factors and irreducible quadratic factors,
each with real coefficients.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example Factoring a Polynomial
Write f ( x)  3x5  x 4  24 x3  8 x 2  27 x  9as a product
of linear and irreducible quadratic factors, each with
real coefficients.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 2- 27
Polynomial Functions of Odd Degree
Every polynomial of odd degree with real coefficients
has at least one real zero.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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