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