1. Solve by Factoring:

2. Solve by completing the Square:

3. Rational Root Test

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

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

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

7. Rational Zero Test If f(x)=a n x n + a n-1 x n-1 +… + a 1 x + a 0 Then the possible rational roots are Factors of the last term (a 0 ) over the factors of the first term (a n )

8. Example

9. Find all real roots: Mult. of 2 Touches. Goes Through xy 01

10. Find all real roots:

11. Goes Through ALL xy 03

12. Find all real roots:

13. xy 0-6 All Go Through

14. Find all real roots:

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

16. Find all real roots:

17. Find all real roots: Do NOT Graph.

18. Find all real roots:

20. Complex Numbers

21. Imaginary Unit (i) =

22. 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

23. Simplify: