1) Find f(g(x)) and g(f(x) to show that f(x) and g(x) are inverses

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Presentation transcript:

Section 2.5: Complex Zeros and Fundamental Theorem of Algebra Section 2.6: Rational Functions 1) Find f(g(x)) and g(f(x) to show that f(x) and g(x) are inverses 2)Find all the rational zeroes of g(x) = 3x3 – 5x2 - 9x + 15 3)Show that all real roots of the equation lie between - 4 and 4.

Review of Imaginary Numbers “i” is equal to . Then, . Complex numbers are a combination of real and imaginary parts: a + bi When adding or subtracting complex numbers, combine like parts. (2 – 3i) + (-7 + 11i) b) (4-6i) – (-8 + 9i) When multiplying complex numbers, FOIL. a) (4 + 5i)(10 + 3i) b) (2 – 5i)(2 + 5i)

Complex Conjugates a + bi and a – bi If a polynomial has a complex zero, then it has at least one more complex zero, its complex conjugate. Complex zeros ALWAYS appear in pairs. If a polynomial of degree 3 has -5 and 4i as zeros, what is the third zero? What is the equation of this polynomial?

More w/ Complex Conjugates If a polynomial of degree 3 has 7 and 5 - 2i as zeros, what is the third zero? What is the equation of this polynomial?

Determining The Number of Real and Complex Zeros Find all real zeros using your graphing calculator. Find all complex zeros by using real zeros in synthetic division to simplify the polynomial. Then, solve for complex zeros. Determine the number of real and complex zeros in the polynomial function . What are the zeros?

Linear Factorization ***factored form*** Write the linear factorization of .

Factorization w/ Irreducible Quadratic Factors Instead of including linear factors with complex zeros, sometimes you will be asked to write your answer as a product of linear factors and irreducible quadratic factors, all with real coefficients. Write as a product of linear factors and irreducible quadratic factors, all with real coefficients.

Given a zero, determine the remaining zeros. Given 5i is a zero of , find all remaining zeros.