1. Solve by Factoring:

2. Solve by completing the Square:

3. Complex Numbers: Consists of a real number plus an imaginary number
Looks like: a + bi
Can also be called an imaginary number
If a = 0, then it’s a pure imaginary number

4. Simplify:

5. Simplify:

6. Simplify:

7. Simplify:

8. Simplify:

9. Simplify:

10. Simplify:

11. Rational Root Test

12. Objective
Use the Rational Root Theorem

13. Objective
Learn how to evaluate data from real world applications that fit into a quadratic model.

14. Remainder Theorem
Remainder = f(k)
Example:
f(2)=

15. Therefore, x = -2 is NOT a root!
Find the remainder:
Therefore, x = -2 is NOT a root!
Factor
Theorem

16. Factor Theorem
f(x) has a factor (x-k) iff f(k)=0.

17. Rational Root Theorem If f(x)=anxn + an-1xn-1 +… + a1x + a0
Then the possible rational roots are
Factors of the last term (a0) over the factors of the first term (an)

18. Example

19. Find all real roots: x y 1
Mult. of 2
Touches.
Goes Through

20. Find all real roots:

21. Find all real roots: x y 3
Goes Through ALL

22. Find all real roots: x y
All Go Through -6

23. Find all real roots:

24. Find all real roots: Do NOT Graph.
NOT Real!

25. Find all real roots: Do NOT Graph.

26. Find all real roots:

27. Find all real roots:

28. Complex Numbers

29. Imaginary Unit (i) =