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

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.

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

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

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 3 +16 𝑥 2 −20𝑥+4 𝑥 2 −16𝑥+20 𝑓 𝑥 = 𝑥 4 −8 𝑥 3 +25 𝑥 2 −36𝑥+20

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+75 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

Complex Zeros; Fundamental Theorem of Algebra 𝑓 𝑥 = 𝑥 3 +3 𝑥 2 +25𝑥+75 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

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, ±20 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

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)

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