PC Unit 6 Lecture 1, 112002

1
PC Unit 6 Lecture 1, 11/20/02

• The Real Zeros of a Polynomial Function (Section
  4.1)
• Remainder and Factor Theorems
• Descartes Rule
• Rational Zeros Theorem
• Complex Numbers (Section 4.2)

Homework Read 4.1-4.3 Problems Sec. 4.1 -
3-5odd, 15-19odd, 25-29odd, 45-49odd, 67 69
Sec. 4.2 - 3-5odd, 7-9odd, 11-13odd, 15-17odd,
25-31odd, 33-37odd, 39-43odd,
45-59odd, 65-67odd, 73-75odd
Some slides may be from the Prentice Hall Website
for the textbook Precalculus Graphing and Data
Analysis by Michael Sullivan Michael Sullivan
III, 2001. (www.prenhall.com/sullivan)
2
Test 5 Statistics

• Average 73 with bonus points
• Standard Deviation 12.5
• Highest Grade 99
• Letter grade distribution
• 3 As
• 6 Bs
• 9 Cs
• 10 Ds
• 17 Fs

3
Test Curve? Test 5 Rework Requirements

• Will there be a curve?
• Yet again!
• Who will get the benefit of a curve?
• Only those who try to help themselves by
  reworking the test!!!!!
• When is the rework due? (Monday, Due 11/25)
• What is required?
• Each problem missed must be reworked. This may
  be done on the test or separate paper. If done
  on the test, make it clear what is changed.
• Dont forget to turn in your test with your
  reworked problems.

4
Algebra Review Assignment, Ch. 5

• Same as before.
• If your 2nd 6-weeks grade was a B- or higher, do
  25 problems from the Ch. 5 Review questions. You
  have the option of doing 4 problems from each
  subsection of sections 5.1-5.7. Each subsection
  must have a percent correct gt80.
• If your grade was C or lower, then you MUST do 4
  problems from each subsection of sections
  5.1-5.7. This will be a total of 92 problems.
  Each subsection must have a percent correct
  gt80.
• For a grade of
• 10 out of 10, your percent correct must be 93
• 9 out of 10, your percent correct must be between
  8392
• 8 out of 10, your percent correct must be between
  80-82.
• 0 out of 10. No credit will be given if your
  percent is lower than 80.

5
Remainder Theorem

• Remainder Theorem Let f be a polynomial
  function. If f(x) is divided by x-c, then the
  remainder is .
• See Example 1, page 240.

6
Example 1
This is a modified slide from the Prentice Hall
website.
7
Factor Theorem

• Factor Theorem Let f be a polynomial function.
  Then x-c is a factor of f(x) if and only if
  .
• If , then x-c is a factor of f(x).
• If x-c is a factor of f(x), then

8
Example 2
(a) x 3 (b) x 4
This is a modified slide from the Prentice Hall
website.
9
Descartes Rule of Signs

• Let f denote a polynomial function.
• Number of positive real zeros - equals of sign
  variations of f(x) or is an even integer number
  less.
• Number of negative real zeros - equals of sign
  variations of f(-x) or is an even integer number
  less.

10
Example 3
Discuss the real zeros of
This is a slide from the Prentice Hall website.
11
Rational Zeros Theorem

• Let f be a polynomial function of degree 1 or
  higher of the form
  where each coefficient
  is an integer. If p/q, in lowest terms, is a
  rational zero of f, then p must be a factor of
  a0, and q must be a factor of an.

12
Example 4
List the potential rational zeros of
This is a slide from the Prentice Hall website.
13
Example 5
Find the real zeros of
Factor f over the reals.
This is a slide from the Prentice Hall website.
14
Example 5 continued
Thus, -3 is a zero of f and x 3 is a factor of
f.
This is a slide from the Prentice Hall website.
15
Example 5 continued
Thus, -2 is a zero of f and x 2 is a factor of
f.
This is a slide from the Prentice Hall website.
16
Section 4.2 Complex Numbers
17
The imaginary unit is denoted i and by definition
its square is 1.
18
Adding and Subtracting Complex Numbers
Adding
(a bi) (c di) (a c) (b d)i
(2 4i) (-1 6i) (2 - 1) (4 6)i
1 10i
Subtracting
(a bi) - (c di) (a - c) (b - d)i
(3 i) - (1 - 2i) (3 - 1) (1 - (-2))i
2 3i
This is a modified slide from the Prentice Hall
website.
19
Multiplying Complex Numbers
Foil and group real terms and the imaginary terms.
This is a modified slide from the Prentice Hall
website.
20
The Complex Conjugate
Theorem
21
Writing a Complex Number Quotient in Standard Form
The complex conjugate is used to get rid of the
imaginary terms in the denominator.
This is a modified slide from the Prentice Hall
website.
22
The Quadratic Formula
This is a slide from the Prentice Hall website.
23
Example 6
This is a slide from the Prentice Hall website.