Download presentation

1
**Complex Zeros; Fundamental Theorem of Algebra**

Objective: SWBAT identify complex zeros of a polynomials by using Conjugate Root Theorem SWBAT find all real and complex zeros by using Fundamental Theorem of Algebra

2
**Complex Zeros; Fundamental Theorem of Algebra**

Complex Numbers The complex number system includes real and imaginary numbers. Standard form of a complex number is: a + bi. a and b are real numbers. i is the imaginary unit −1 ( 𝑖 2 =−1). Fundamental Theorem of Algebra Every complex polynomial function of degree 1 or larger (no negative integers as exponents) has at least one complex zero.

3
**Complex Zeros; Fundamental Theorem of Algebra**

Every complex polynomial function of degree n 1 has exactly n complex zeros, some of which may repeat. Conjugate Pairs Theorem If 𝑟=𝑎+𝑏𝑖 is a zero of a polynomial function whose coefficients are real numbers, then the complex conjugate 𝑟 =𝑎−𝑏𝑖 is also a zero of the function. Examples 1) A polynomial function of degree three has 2 and 3 + i as it zeros. What is the other zero? 𝑥=3−𝑖

4
**Complex Zeros; Fundamental Theorem of Algebra**

Examples 2) A polynomial function of degree 5 has 4, 2 + 3i, and 5i as it zeros. What are the other zeros? 𝑥=2−3𝑖 𝑎𝑛𝑑 𝑥=−5𝑖 3) A polynomial function of degree 4 has 2 with a zero multiplicity of 2 and 2 – i as it zeros. What are the zeros? 𝑥=2 𝑟𝑒𝑝𝑒𝑎𝑡𝑠 𝑡𝑤𝑖𝑐𝑒 𝑎𝑛𝑑 𝑥=2+𝑖

5
**Complex Zeros; Fundamental Theorem of Algebra**

Examples 4) A polynomial function of degree 4 has 2 with a zero multiplicity of 2 and 2 – i as it zeros. What is the function? 𝑥=2 𝑥=2 𝑥=2−𝑖 𝑥=2+𝑖 𝑓 𝑥 =(𝑥−2)(𝑥−2)(𝑥−(2−𝑖))(𝑥−(2+𝑖)) 𝑓 𝑥 = (𝑥 2 −4𝑥+4)(𝑥−2+𝑖)(𝑥−2−𝑖) ( 𝑥 2 −2𝑥−𝑖𝑥−2𝑥+4+2𝑖+𝑖𝑥−2𝑖− 𝑖 2 ) 𝑓 𝑥 = (𝑥 2 −4𝑥+4)( 𝑥 2 −4𝑥+5) 𝑓 𝑥 = 𝑥 4 −4 𝑥 3 +5 𝑥 2 −4 𝑥 𝑥 2 −20𝑥+4 𝑥 2 −16𝑥+20 𝑓 𝑥 = 𝑥 4 −8 𝑥 𝑥 2 −36𝑥+20

6
**Complex Zeros; Fundamental Theorem of Algebra**

Find the remaining complex zeros of the given polynomial functions 5) 𝑓 𝑥 = 𝑥 3 +3 𝑥 2 +25𝑥 𝑧𝑒𝑟𝑜:−5𝑖 𝐴𝑛𝑜𝑡ℎ𝑒𝑟 𝑧𝑒𝑟𝑜 (𝑡ℎ𝑒 𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒):5𝑖 𝑥=−5𝑖 𝑎𝑛𝑑 𝑥=5𝑖 (𝑥+5𝑖)(𝑥−5𝑖) 𝑥 2 −5𝑖𝑥+5𝑖𝑥−25 𝑖 2 𝑥 2 −25(−1) 𝑥 2 +25

7
**Complex Zeros; Fundamental Theorem of Algebra**

𝑓 𝑥 = 𝑥 3 +3 𝑥 2 +25𝑥 𝑧𝑒𝑟𝑜:−5𝑖 Long Division 𝑥 +3 𝑥 3 25𝑥 3𝑥 2 +75 3𝑥 2 +75 𝑥+3 𝑥 2 +25 (𝑥+3)(𝑥+5𝑖)(𝑥−5𝑖) 𝑧𝑒𝑟𝑜𝑠:−3,−5𝑖 𝑎𝑛𝑑 5𝑖

8
**Complex Zeros; Fundamental Theorem of Algebra**

Find the complex zeros of the given polynomial functions 6) 𝑓 𝑥 = 𝑥 4 −4 𝑥 3 +9 𝑥 2 −20𝑥+20 𝑝: ±1, ±2, ±4, ±5, ±10, ± 𝑞: ±1 𝑝 𝑞 : ± 1 1 , ± 2 1 , ± 4 1 , ± 5 1 , ± 10 1 , ± 20 1 Possible solutions: 𝑥=±1, ±2, ±4, ±5,±10,±20 Try: 𝑥=1 Try: 𝑥=−1 1 −3 6 −14 −1 5 −14 34 1 −3 6 −14 6 1 −5 14 −34 54

9
**Complex Zeros; Fundamental Theorem of Algebra**

𝑓 𝑥 = 𝑥 4 −4 𝑥 3 +9 𝑥 2 −20𝑥+20 Try: 𝑥=2 2 −4 10 −20 1 −2 5 −10 𝑓 𝑥 =(𝑥−2)( 𝑥 3 −2 𝑥 2 +5𝑥−10) 𝑓 𝑥 =(𝑥−2)( 𝑥 2 𝑥−2 +5(𝑥−2)) 𝑓 𝑥 =(𝑥−2)(𝑥−2)( 𝑥 2 +5)

10
**Complex Zeros; Fundamental Theorem of Algebra**

𝑓 𝑥 = 𝑥 4 −4 𝑥 3 +9 𝑥 2 −20𝑥+20 𝑓 𝑥 = 𝑥−2 𝑥−2 𝑥 2 +5 =0 𝑥−2=0 𝑥−2=0 𝑥 2 +5=0 𝑥 2 =−5 𝑥=2 𝑥=± −5 𝑧𝑒𝑟𝑜 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 2 𝑥=± 5 𝑖 Complex zeros: 2 with multiplicity of 2, 5 𝑖, 𝑎𝑛𝑑 − 5 𝑖 𝑓 𝑥 𝑖𝑛 𝑓𝑎𝑐𝑡𝑜𝑟𝑒𝑑 𝑓𝑜𝑟𝑚 𝑓 𝑥 = (𝑥−2) 2 𝑥− 5 𝑖 𝑥+ 5 𝑖

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google