Pre-AP Pre-Calculus Chapter 2, Section 5 Complex Zeros and the fundamental Theorem of Algebra

Use your device to answer the following questions…. What is an imaginary number? How do you represent an imaginary number? Why do we need imaginary numbers? What is a complex number?

Use your device…. Sketch an example of a graph that has no real zeros.

Theorem: Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

Something to understand…. Just because you can find a nonreal zero of a function doesn’t mean the graph touches or crosses the x-axis.

Write the polynomial in standard form, and identify the zeros of the function and the x-intercepts of the graph. 𝑓 𝑥 =(𝑥−2𝑖)(𝑥+2𝑖)

Write the polynomial in standard form, and identify the zeros of the function and the x-intercepts of the graph. 𝑓 𝑥 =(𝑥−5)(𝑥− 2 𝑖)(𝑥+ 2 𝑖)

Write the polynomial in standard form, and identify the zeros of the function and the x-intercepts of the graph. 𝑓 𝑥 =(𝑥−3)(𝑥−3)(𝑥−𝑖)(𝑥+𝑖)

Theorem: Complex Conjugate Zeros Suppose that f(x) is a polynomial function with real coefficients. If a and b are real numbers with 𝑏≠0 and a + bi is a zero of f(x), then its complex conjugate a – bi is also a zero of f(x). What does that mean?? If a function has a complex number as a nonreal zero, then its conjugate is also a part of function. This is used because when you multiply a complex number by its conjugate, you will get a real number.

Complex conjugates Expression Complex conjugate 𝑧 3+2𝑖 3−2𝑖 5𝑖 −5𝑖 4+𝑖 4−𝑖 7

Finding a Polynomial from Given Zeros Write a polynomial function of minimum degree in standard form whose zeros include -3, 4, and 2 + i.

Finding a Polynomial from Given Zeros Write a polynomial function of minimum degree in standard form whose zeros include 𝑥=1, 𝑥=1+2𝑖, and 𝑥=1−𝑖.

Factoring a Polynomial with Complex Zeros Find all zeros of 𝑓 𝑥 = 𝑥 5 −3 𝑥 4 −5 𝑥 3 +5 𝑥 2 −6𝑥+8, and write 𝑓(𝑥) in its linear factorization. Hint: find all real zeros using the calculator first.

Finding Complex Zeros The complex number 𝑧=1−2𝑖 is a zero of 𝑓 𝑥 =4 𝑥 4 +17 𝑥 2 +14𝑥+65. Find the remaining zeros of 𝑓(𝑥), and write it in its linear factorization.

Practice Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicities include those listed. -1 (multiplicity 3), 3 (multiplicity 1)

Practice Find all of the zeros and write a linear factorization of the function

𝑓 𝑥 = 3𝑥 5 − 2𝑥 4 + 6𝑥 3 −4 𝑥 2 −24𝑥+16 Find all the possible zeros of the function and determine which ones are the zeros. Factor completely.

Ch. 2.5 Homework Pg. 234 – 235, #’s: 3, 9, 15, 23, 25 27, 29, 35, 39, 43, 49 11 Total Problems Gray Book: 215 - 216