1. Homework Log Thurs 11/19 Lesson 5 – 1 Learning Objective:
To classify polynomials
Hw: Pg.

2. 11/19/15 Lesson 5 – 1 Polynomial Functions
Algebra II

3. Learning Objective
To classify polynomials

4. Polynomial Degree Name1 2 3 4 5
Constant
Linear
Quadratic
Cubic
Quartic
Quintic
5 x 𝑥 2 4 𝑥 3 −2 𝑥 2 +𝑥 2 𝑥 4 +5 𝑥 2 − 𝑥 5 +4 𝑥 2 +2𝑥+1

5. Polynomial # of terms Name1 2 3 4
Monomial
Binomial
Trinomial
Polynomial
5 x 𝑥 2 4 𝑥 3 −2 𝑥 2 +𝑥 2 𝑥 4 +5 𝑥 2 − 𝑥 5 +4 𝑥 2 +2𝑥+1

6. Standard Form of a Polynomial Function
Arrange the terms by degree in descending numerical order.
Write in standard form & classify by degrees & number of terms.
1. 3x+9 𝑥 2 +5
9 𝑥 2 +3𝑥+5
Quadratic Trinomial

7. Write in standard form & classify by degrees & number of terms
2. 4x−6 𝑥 2 + 𝑥 𝑥 2 −12 𝑥 4 +4 𝑥 2 +4𝑥−12 Quartic Polynomial 3. 4 𝑥 3 − 3𝑥 5 − 3𝑥 5 +4 𝑥 3 Quintic Binomial

8. End Behavior
+ 𝑥 𝑒𝑣𝑒𝑛 − 𝑥 𝑒𝑣𝑒𝑛
+ 𝑥 𝑜𝑑𝑑
− 𝑥 𝑜𝑑𝑑

9. 11/19/15 Lesson 5 – 2 Polynomials, Linear Factors, & Zeros
Algebra II

10. Learning Objective To analyze the factored form of a polynomial
To write a polynomial function from its zeros

11. –15
Factor & Find Zeros
1. 𝑥 3 −2 𝑥 2 −15𝑥 x 𝑥 2 −2𝑥−15 x 𝑥 2 −5𝑥+3𝑥−15 x(x(x – 5) + 3(x – 5)) x(x + 3)(x – 5) Find zeros by = 0 x(x + 3)(x – 5) = 0 x = 0, –3, 5 –5 3 –2

12. –12
Factor & Find Zeros
2. 𝑥 3 − 𝑥 2 −12𝑥 x 𝑥 2 −𝑥−12 x 𝑥 2 −4𝑥+3𝑥−12 x(x(x – 4) + 3(x – 4)) x(x + 3)(x – 4) Find zeros by = 0 x(x + 3)(x – 5) = 0 x = 0, –3, 4 –4 3 –1

13. Write a Polynomial Function From its Zeros
3. What is a cubic polynomial function in standard form with zeros –2, 2, and 3? x = –2 x = 2 x = 3 x + 2 = 0 x – 2 = 0 x – 3 = 0 (x + 2)(x – 2)(x – 3) = 0

14. (x + 2)(x – 2)(x – 3) = 0 (𝑥 2 −4)(x−3) x – 2 x – 3 x 𝑥 2 –2x 𝑥 2 𝑥 3
–3 𝑥 2 2 2x –4 –4
–4x 12
(𝑥 2 −4)
𝑥 3 −3 𝑥 2 −4𝑥+12=0

15. Write a Polynomial Function From its Zeros
3. What is a cubic polynomial function in standard form with zeros –2, 2, and 3? x = –2 x = 2 x = 3 x + 2 = 0 x – 2 = 0 x – 3 = 0 (x + 2)(x – 2)(x – 3) = 0 𝑥 2 −4 𝑥−3 =0 𝑥 3 −3 𝑥 2 −4𝑥+12=0

16. Write a Polynomial Function From its Zeros
4. What is a cubic polynomial function in standard form with zeros 0, 5, and 5? x = 0 x = 5 x = 5 x = 0 x – 5 = 0 x – 5 = 0 (x)(x – 5)(x – 5) = 0 (𝑥 2 −5𝑥)(𝑥−5)=0 𝑥 3 −10 𝑥 2 +25𝑥=0

17. Write a Polynomial Function From its Zeros
5. What is a quartic polynomial function in standard form with zeros –2, –2, 2 and 3? x = –2 x = –2 x = 2 x = 3 x+2=0 x+2=0 x–2=0 x-3=0 (x + 2)(x + 2)(x – 2)(x - 3)=0

18. (x + 2)(x + 2)(x – 2)(x - 3)=0 x 2 x – 3 x 𝑥 2 2x x 𝑥 2 –3x 2 –2 2x 46
(𝑥 2 +4x+4)
(𝑥 2 −5𝑥+6)

19. (𝑥 2 +4x+4) (𝑥 2 −5𝑥+6)=0 -5𝑥 𝑥 2 6 𝑥 2 6 𝑥 2 𝑥 4 -5 𝑥 3 4x 4 𝑥 3
–20 𝑥 2
24x 4
4 𝑥 2
-20x 24
𝑥 4 − 𝑥 3 −10 𝑥 2 +4𝑥+24=0

20. Write a Polynomial Function From its Zeros
5. What is a quartic polynomial function in standard form with zeros –2, –2, 2 and 3? x = –2 x = –2 x = 2 x = 3 x+2=0 x+2=0 x–2=0 x-3=0 (x + 2)(x + 2)(x – 2)(x - 3)=0 𝑥 4 − 𝑥 3 −10 𝑥 2 +4𝑥+24=0

21. Multiplicity of a Zero Multiplicity – how many times a factor appears
Even multiplicity – TOUCHES the x-axis, will NOT cross
Odd multiplicity – CROSSES the x-axis

22. Multiplicity of a Zero
6. What are the zeros of f x = 𝑥 4 −2 𝑥 3 −8 𝑥 2 ? What are the multiplicities? 𝑓 𝑥 =𝑥 2 ( 𝑥 2 −2𝑥−8) 𝑓 𝑥 =𝑥 2 (𝑥−4)(𝑥+2) Zeros: 𝑥 2 𝑥−4 𝑥+2 =0 x = 0 (multiplicity 2) x = 4 (multiplicity 1) x = –2 (multiplicity 1)

23. Multiplicity of a Zero
7. What are the zeros of f x = 𝑥(𝑥−3) 4 (𝑥+3) 3 (𝑥+5) 2 ? What are the multiplicities? x = 0 (multiplicity 1) x = 3 (multiplicity 4) x = –3 (multiplicity 3) x = –5 (multiplicity 2)

24. Ticket Out the Door
Write a cubic polynomial function in standard form with roots 4, 2, and – 1.

25. Assignment: Pg. WS 5.1 and 5.2