Homework Log Thurs 11/19 Lesson 5 – 1 Learning Objective: To classify polynomials Hw: 5.1 – 5.2 WS

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

Learning Objective To classify polynomials

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

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

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

Write in standard form & classify by degrees & number of terms 2. 4x−6 𝑥 2 + 𝑥 4 +10 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

End Behavior + 𝑥 𝑒𝑣𝑒𝑛 − 𝑥 𝑒𝑣𝑒𝑛 + 𝑥 𝑜𝑑𝑑 − 𝑥 𝑜𝑑𝑑

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

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

Factor & Find Zeros –15 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

Factor & Find Zeros –12 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

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)(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

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

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

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

(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

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

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

Multiplicity of a Zero 6. 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)

Multiplicity of a Zero 7. 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)

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

Assignment: Pg. 5.1 – 5.2 WS