4C - I can explain how the Division Algorithm, Remainder ... · 4.2 Polynomials v1 20131118 (period...

4.2 Polynomials v1 20131118 (period 2).notebook

1

November 19, 2013

Oct 15­8:53 AM

Task 4.2 - Polynomial Functions

Learning Targets:

4A - I can describe a polynomial function in standard form and classify special cases of polynomials.

4B - I can apply the Fundamental Theorem of Algebra and connect to finding solutions to polynomial functions.

4C - I can explain how the Division Algorithm, Remainder Theorem, and Factor Theorem are related, and their purpose in finding solutions.

4.2 Polynomials v1 20131118 (period 2).notebook

2

November 19, 2013

Sep 16­10:50 AM

I) Polynomials in Standard FormA) Vocabulary

1) Polynomial - A monomial or sum of monomials

2) Degree - Highest exponent

4.2 Polynomials v1 20131118 (period 2).notebook

3

November 19, 2013

Sep 16­10:50 AM

3) Standard Form - List terms in descending order by degree.

4) Multiplicity - The number of times a root is a root

4.2 Polynomials v1 20131118 (period 2).notebook

4

November 19, 2013

Sep 16­10:50 AM

II) Fundamental Theorem of AlgebraA) Every polynomial having complex coefficients

and degree ≥ 1, has at least 1 complex root

*Every non-zero polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity

4.2 Polynomials v1 20131118 (period 2).notebook

5

November 19, 2013

Sep 16­10:50 AM

B) Examples - How many roots? Apply FTA1)

2)

4.2 Polynomials v1 20131118 (period 2).notebook

6

November 19, 2013

Nov 19­2:07 PM

4.2 Polynomials v1 20131118 (period 2).notebook

7

November 19, 2013

Sep 16­10:50 AM

III) Roots of polynomialsA) Previously Learned

1)_____, ______, _______, _______ mean the same thing.

4.2 Polynomials v1 20131118 (period 2).notebook

8

November 19, 2013

Sep 16­10:50 AM

2) Factoring Methodsa) b)c)d)e)f)

4.2 Polynomials v1 20131118 (period 2).notebook

9

November 19, 2013

Sep 16­10:50 AM

3) EX-Find the zeros of the polynomial functions, and their multiplicity. How many are real and how many are imaginary?

a)

4.2 Polynomials v1 20131118 (period 2).notebook

10

November 19, 2013

Sep 16­10:50 AM

b)

4.2 Polynomials v1 20131118 (period 2).notebook

11

November 19, 2013

Sep 16­10:50 AM

Practice - Work on "The Fundamental Theorem of Algebra" #1-8

4.2 Polynomials v1 20131118 (period 2).notebook

12

November 19, 2013

Sep 16­10:50 AM

B) Division Algorithm1) Given a numerator (dividend) and denominator

(divisor), an algorithm that computes their quotient and/or remainder

4.2 Polynomials v1 20131118 (period 2).notebook

13

November 19, 2013

Sep 16­10:50 AM

B) Division Algorithm

P(x) = D(x) Q(x) + R(x)

4.2 Polynomials v1 20131118 (period 2).notebook

14

November 19, 2013

Sep 16­10:50 AM

2) Prior- Divide and write in the form:dividend = divisor * quotient + remainder

4.2 Polynomials v1 20131118 (period 2).notebook

15

November 19, 2013

Sep 16­10:50 AM

Your Turn...Work on #9-13 in packet

4.2 Polynomials v1 20131118 (period 2).notebook

16

November 19, 2013

Sep 16­10:50 AM

3) New - Divide using Long Division and write in the form:P(x) = D(x) Q(x) + R(x)

4.2 Polynomials v1 20131118 (period 2).notebook

17

November 19, 2013

Sep 16­10:50 AM

Now your turn...work on #14-16

4.2 Polynomials v1 20131118 (period 2).notebook

18

November 19, 2013

Sep 16­10:50 AM

3) New - Divide using Sythetic Division & write in form:P(x) = D(x) Q(x) + R(x)

4.2 Polynomials v1 20131118 (period 2).notebook

19

November 19, 2013

Sep 16­10:50 AM

Now your turn...work on #17-19

4.2 Polynomials v1 20131118 (period 2).notebook

20

November 19, 2013

Sep 16­10:50 AM

C) The Remainder Theorem1) If P(x) is a polynomial, and "r" is a real number,

then if P(x) is divided by (x-r), then the remainder "R" is P(r).

4.2 Polynomials v1 20131118 (period 2).notebook

21

November 19, 2013

Sep 16­10:50 AM

2) Example: P(x) = x3+ 2x2+3x- 1

4.2 Polynomials v1 20131118 (period 2).notebook

22

November 19, 2013

Sep 16­10:50 AM

3) Think! Why would P(r)=0 be important?

4.2 Polynomials v1 20131118 (period 2).notebook

23

November 19, 2013

Sep 16­10:50 AM

3) Think! Why would P(r)=0 be important?

It means that x=r is a root/solution, and (x-r) is a factor.

4.2 Polynomials v1 20131118 (period 2).notebook

24

November 19, 2013

Sep 16­10:50 AM

D) Factor Theorem1) If P(x) is a polynomial function, then (x-r) is a

factor of P(x) if and only if R=0.

3) Three methods to determine Remainder R:a) Long Divisionb) Synthetic Division (linear only)c) Remainder Theorem

4.2 Polynomials v1 20131118 (period 2).notebook

25

November 19, 2013

Sep 16­10:50 AM

2) Example: P(x)= 2x3- x2+ 2x - 3a) Is (x-1) a factor of P(x)?

b) Is (x+2) a factor of P(x)?