1. Rational Root and Complex Conjugates Theorem

2. Rational Root Theorem
Used to find possible rational roots (solutions) to a polynomial
Possible Roots : P/Q
Where P represents the factors of the constant of the polynomial and Q represents the factors of the leading coefficient

3. Rational Root Theorem

4. Another example

5. List all possible rational roots.
𝑓 𝑥 =12 𝑥 5 +6 𝑥 4 −7 𝑥 3 +2 𝑥 2 −3𝑥+18

6. Find all possible rational roots and then find ones that work.

7. Rational Root Theorem

8. Conjugate Root Theorem
If 𝑃(𝑥) is a polynomial with real coefficients, then irrational roots of 𝑃(𝑥) that have the form 𝑎+ 𝑏 occur in conjugate pairs. That is, if 𝑎+ 𝑏 is an irrational root, then 𝑎− 𝑏 is also a root. Likewise for complex roots, if 𝑎+𝑏𝑖 is a root, then 𝑎−𝑏𝑖 is also a root.

9. Find the lowest degree polynomial with the following roots
3 and 2 – i.

10. Find the lowest degree polynomial with the following roots

11. Find the lowest degree polynomial with the following roots