1. Warm Up – Look over your notes/examples from last week Any questions??

2. Pop Quiz! – Clear your desk except for a pencil & calculator!

3. Guided Notes – complete from Friday

4. 6.5 Theorems About Roots

5. The Degree
REMEMBER: the DEGREE of a polynomial is the highest exponent! The Degree also tells us the number of Solutions! Including Real AND Imaginary!

6. Irrational Root Theorem and Imaginary Root Theorem
Irrational Root Theorem Square Root Solutions come in PAIRS! If x2 = c then x = ± √c If √ is a solution so is -√ Imaginary Root Theorem If a + bi is a solution, so is a – bi

7. Get ready!
Solve the following by taking the square root. X2 – 49 = 0 X = 0

8. Using the Theorems √5 2. 4i 3. 2 – i 4. 2 - √3
Given one Root, find the other root!
√ i
3. 2 – i √3

9. Zeros to Factors
If a is a zero, then (x – a) is a factor!!

10. Examples
Find a 2nd degree equation with roots 2 and 3 Find a 2nd degree equation with roots -1 and 6

11. Examples
Find a 2nd degree equation with roots 0 and 4 Find a 2nd degree equation with roots ±√7

12. Examples
Find a 2nd degree equation with roots ±2√5 Find a 2nd degree equation with roots ±6i

13. Examples
Find a 2nd degree equation with roots 2 – 3i and 2 +3i

14. Find a 2nd degree equation with a root of 7 + i

15. Examples
Find a 3rd degree equation with roots 4 and 3i

16. Extra Practice – Solving
How many solutions will each equation have? Find them!
x3 – 6x2 – 16x = 0
X = 0

17. Solving by Graphing
Keep in mind, Solving by Graphing ONLY works for REAL SOLUTIONS. You cannot find Imaginary solutions from a Graph. Roots: This is another word for zeros or solutions.

18. Example: Solve
x3 – 5x2 - 2x + 24 = 0

19. Example
24x3 – 22x2 - 5x + 6 = 0

20. Solve x3 – 5x2 + 7x – 35 = 0