Complex Zeros; Fundamental Theorem of Algebra

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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.

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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 (

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

Theorem

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

Conjugate Pairs Theorem

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

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?
• 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?

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?

Complex Zeros; Fundamental Theorem of Algebra

Find the remaining complex zeros of the given polynomial functions

5)

Complex Zeros; Fundamental Theorem of Algebra

Find the complex zeros of the given polynomial functions

6)

Possible solutions:

Try:

Try: