Polynomials and Polynomial Functions

1
Chapter 6

• Polynomials and Polynomial Functions

2
Instructions for this PowerPoint.

• If there is a linked word(s), then click on them.
• If there is an arrow in the lower corner, then
  follow it after you have used all links.
• If there are no links or arrows then just click
  to proceed to the next slide.

3
Polynomial Functions

• Exploring Polynomial Functions
• Examples
• Modeling Data with Polynomial Functions
• Examples

4
Degree Name of Degree Number of Terms Name using number of terms
0 Constant 1 Monomial
1 Linear 2 Binomial
2 Quadratic 3 Trinomial
3 Cubic 4 3rd degree Polynomial
4 Quartic n 4th degree Polynomial with n terms
5 Quintic 5th degree polynomial with n terms
5
x 0 3 5 6 9 11 12 14
y 42 31 26 21 17 15 19 22
6
(No Transcript)
7
Polynomials and Linear Factors

• Standard Form
• Example
• Factored Form
• Examples
• Factors and Zeros
• Examples

8
Writing a polynomial in standard form
You must multiply
(x 1)(x2)(x3)
X3 6x2 11x 6
9
2x3 10x2 12x
2x(x2 5x 6)
10
Factor Theorem
The expression x-a is a linear factor of a
polynomial if and only if the value a is a zero
of the related polynomial function.
11
Factors and Zeros
ZEROS
FACTORS

• -3
• -2
• -1
• 0
• 1
• 2
• 3

• (x (-3)) or (x 3)
• (x (-2)) or (x 2)
• (x (-1)) or (x 1)
• (x 0) or x
• (x 1)
• (x 2)
• (x 3)

12
Dividing Polynomials

• Long Division
• Synthetic Division

13
Long Division

• The purpose of this type of division is to use
  one factor to find another.

)
4
40
x - 1
x3 6x2 -6x - 1
Just as 4 finds the 10
The (x-1) finds the (x2 7x 1)
14
Synthetic Division

• When dividing by x a, use synthetic division.
• The Remainder Theorem

15
The Remainder Theorem

• When using Synthetic Division, the remainder is
  the value of f(a).
• This method is as good as PLUGGING IN, but may
  be faster.

16
Solving Polynomial Equations

• Solving by Graphing
• Solving by Factoring

17
Solving by Graphing

• Set equation equal to 0, then substitute y for 0.
  Look at the x-intercepts. (Zeros)
• Let the left side be y1and let the right side be
  y2. (Very much like solving a system of
  equations by graphing). Look at the points of
  intersection.

18
Solving by Factoring

• Sum of two cubes
• (a3 b3) (a b)(a2 ab b2)
• Difference of two cubes
• (a3 b3) (a b)(a2 ab b2)

19
More on Factoring

• If a polynomial can be factored into linear or
  quadratic factors, then it can be solved using
  techniques learned from earlier chapters.
• Solving a polynomial of degrees higher than 2 can
  be achieved by factoring.

20
Theorems about Roots

• Rational Root Theorem
• Irrational Root Theorem
• Imaginary Root Theorem

21
Rational Root Theorem

• What are Rational Roots?
• Ps and Qs . )
• Using the calculator to speed up the process.

22
And the Rational Roots are..
P includes all of the factors of the constant.
Q includes all of the factors of the leading
coefficient.
f(x) x3 13x - 12
The possible rational roots are
p 12 q 1
23
Test the Possible Roots
In this case all roots are real and rational,
but you need only to find one rational root.
This will become clear later.
24
Since -1, -3, and 4 are the Roots, (x 1), (x
3), and (x 4) are the factors.
Multiply to show that
(x1)(x3)(x-4) x3 13x 12
(x1)(x2 x 12) x3 x2 12x
x2 x 12 x3 13x 12

25
Irrational Root Theorem
These are called CONJUGATES.
26
Imaginary Root Theorem
These are called CONJUGATES.
27
The Fundamental Theorem of Algebra

• If P(x) is a polynomial of degree
  with complex coefficients, then P(x) 0 has
  at least one complex root.
• A polynomial equation with degree n will have
  exactly n roots the related polynomial function
  will have exactly n zeros.

28
The Binomial Theorem

• Binomial Expansion and Pascals Triangle
• The Binomial Theorem

29
PASCALS TRIANGLE
1 1 1 1 2 1 1 3 3 1 1
4 6 4 1 1 5 10 10 5
1 1 6 15 20 15 6 1 1 7
21 35 35 21 7 1 1 8 28
56 70 56 28 8 1 1 9 36
84 126 126 84 36 9 1