1. 6.5 Theorems About Roots of Polynomial Equations Imaginary Roots

2. POLYNOMIALS and THEOREMS Theorems of Polynomial Equations There are 4 BIG Theorems to know about Polynomials 1)Rational Root Theorem 2)Irrational Root Theorem 3)Imaginary Root Theorem 4)Descartes Rule

3. 1. For polynomial has roots 5 and 3i Other roots ______ - 3i Degree of Polynomial ______ 3 2. For polynomial has roots -5i, 4 – 2i Other roots __________ 5i, 4 + 2i Degree of Polynomial ______ 4 Why not -5 ? For real numbers we do not need to worry about the conjugate. Only the complex part has a second root. Why a third degree polynomial? (x-3i)(x+3i)(x-5)

4. Write a polynomial given the roots 4, 2i, -√5 Other roots are -2i and √5 Put in factored form y = (x – 4)(x + 2i)(x – 2i)(x + √5)(x – √5) Decide what to FOIL first

5. X +√5 x -√5 X2X2 X √5 -X √5 -5 (x² – 5) x 2i x -2i X2X2 2X i -2X i -4i 2 = 4 (x² +4) y = (x – 4)(x + 2i)(x – 2i) (x + √5)(x – √5)

6. y = (x – 4)(x 4 – 5x² + 4x² – 20) y = (x – 4)(x 4 – x² – 20) BOX it!!! x 4 -x 2 -20 x -4 X5X5 -X 3 -4X 4 4x 2 y = x 5 – 4x 4 – x 3 + 4x² – 20x + 80 x 2 4 x 2 -5 X4X4 4x 2 -5X 2 -20 (x 4 – 5x² + 4x² – 20) y = (x – 4)(x 4 – x² – 20) y = (x 4 – x² – 20) -20X 80

7. Write a polynomial given the roots 1 + i Other root is 1 – i Put in factored form y = (x – (1 + i))(x – (1 – i)) Distribute the negatives y = (x – 1 – i )(x – 1 + i) FOIL or BOX to finish it up y = x² – 2x + 2 X²-x-ix - x1 i ix-i-i²=1 X –1 –i X i Note: When you box or foil complex numbers, you will not have any imaginary numbers left in the answer. If you do, go back and check your arithmetic.

8. Let’s Try One Find a fourth-degree polynomial equation with integer coefficients that has the given numbers as roots: 3 + i and -2i (Note: this is problem #29 on tonight’s homework)