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

Standard form of a complex number is: a + bi.

Every complex polynomial function of degree 1 or larger (no negative integers as exponents) has at least one complex zero.

a and b are real numbers.i is the imaginary unit (

The complex number system includes real and imaginary numbers.

Fundamental Theorem of Algebra

Complex Zeros; Fundamental Theorem of Algebra

Conjugate Pairs Theorem

Every complex polynomial function of degree n 1 has exactly n complex zeros, some of which may repeat.

1) A polynomial function of degree three has 2 and 3 + i as it zeros. What is the other zero?

𝑥=3−𝑖

If is a zero of a polynomial function whose coefficients are real numbers, then the complex conjugate is also a zero of the function.

Theorem

Examples

Complex Zeros; Fundamental Theorem of Algebra

2) A polynomial function of degree 5 has 4, 2 + 3i, and 5i as it zeros. What are the other zeros?

𝑥=2−3 𝑖

Examples

𝑥=−5 𝑖𝑎𝑛𝑑3) 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=2𝑟𝑒𝑝𝑒𝑎𝑡𝑠𝑡𝑤𝑖𝑐𝑒 𝑥=2+𝑖𝑎𝑛𝑑

Complex Zeros; Fundamental Theorem of Algebra

𝑥=2−𝑖

Examples4) A polynomial function of degree 4 has 2 with a zero multiplicity of 2 and 2 – i as it zeros. What is the function?

𝑓 (𝑥 )=(𝑥−2)(𝑥−2)(𝑥−(2−𝑖))(𝑥−(2+𝑖))𝑓 (𝑥 )=(𝑥¿¿2−4 𝑥+4 )(𝑥−2+𝑖)(𝑥−2− 𝑖)¿

𝑥=2+𝑖𝑥=2 𝑥=2

(𝑥2−2𝑥− 𝑖𝑥−2 𝑥+4+2 𝑖+𝑖𝑥−2𝑖−𝑖2)𝑓 (𝑥 )=(𝑥¿¿2−4 𝑥+4 )(𝑥2−4 𝑥+5)¿𝑓 (𝑥 )=𝑥4−4 𝑥3+5 𝑥2−4 𝑥3+16 𝑥2−20𝑥+4 𝑥2−16 𝑥+20

𝑓 (𝑥 )=𝑥4−8 𝑥3+25𝑥2−36 𝑥+20

Complex Zeros; Fundamental Theorem of Algebra

Find the remaining complex zeros of the given polynomial functions5)

(𝑥+5 𝑖)(𝑥−5 𝑖)𝑥2−5 𝑖𝑥+5 𝑖𝑥−25 𝑖2

h𝐴𝑛𝑜𝑡 𝑒𝑟 𝑧𝑒𝑟𝑜 ( h𝑡 𝑒𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒) :5 𝑖𝑥=−5 𝑖𝑎𝑛𝑑 𝑥=5 𝑖

𝑥2−25 (−1)𝑥2+25

Complex Zeros; Fundamental Theorem of Algebra

𝑓 (𝑥 )=𝑥3+3𝑥2+25 𝑥+75𝑧𝑒𝑟𝑜 :−5 𝑖

(𝑥+3)(𝑥+5 𝑖)(𝑥−5 𝑖)𝑧𝑒𝑟𝑜𝑠 :−3 ,−5 𝑖𝑎𝑛𝑑5 𝑖

Long Division

𝑥7525325 232 xxxx

𝑥3 25 𝑥3 𝑥2 +75

+3

3 𝑥2 +750

(𝑥+3 ) (𝑥2+25 )

Complex Zeros; Fundamental Theorem of Algebra

Find the complex zeros of the given polynomial functions6)

𝑝 :±1 , ±2 , ±4 ,±5 , ±10 , ±20𝑞 :±1

Possible solutions: Try:

𝑝𝑞:±11,±21, ±41, ±51, ±101, ±201

1

20209411 1−3

−366

6−14

−14

Try:

1

20209411 −1−55

5 414−14−34

34

Complex Zeros; Fundamental Theorem of Algebra

𝑓 (𝑥 )=𝑥4−4 𝑥3+9 𝑥2−20𝑥+20Try:

1

20209412 2−2−4

0510−10

−20

𝑓 (𝑥 )=(𝑥−2)(𝑥3−2 𝑥2+5 𝑥−10)

𝑓 (𝑥 )=(𝑥−2)(𝑥2 (𝑥−2 )+5 (𝑥−2))

𝑓 (𝑥 )=(𝑥−2)(𝑥−2)(𝑥2+5)

Complex Zeros; Fundamental Theorem of Algebra

𝑓 (𝑥 )=𝑥4−4 𝑥3+9 𝑥2−20𝑥+20

𝑓 (𝑥 )=(𝑥−2 ) (𝑥−2 ) (𝑥2+5 )=0𝑥−2=0 𝑥2+5=0𝑥−2=0

𝑥2=−5𝑥=±√−5𝑥=±√5 𝑖

𝑥=2𝑧𝑒𝑟𝑜𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 2

Complex zeros:

𝑓 (𝑥 )=(𝑥−2)2 (𝑥−√5 𝑖 ) (𝑥+√5 𝑖 )𝑓 (𝑥 ) 𝑖𝑛 𝑓𝑎𝑐𝑡𝑜𝑟𝑒𝑑 𝑓𝑜𝑟𝑚