1. Rational Root Theorem Pt.2

2. Learning Targets Introduce formal definition and notation of RRT
Determine all of the possible rational roots for a polynomial
Use the rational root theorem to find a polynomials standard form

3. Formal Definition
If a polynomial function, written in standard form, has integer coefficients, then any rational zero must be of the form ± p/ q, where p is a factor of the constant term and q is a factor of the leading coefficient.

4. From now on:
We will be using the following notation:
± 𝐹𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐹𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝐿𝑒𝑎𝑑𝑖𝑛𝑔 𝐶𝑜𝑒𝑓𝑓𝑒𝑐𝑖𝑒𝑛𝑡 =± 𝑝 𝑞

5. Example:
Determine the possible list of rational roots for the following polynomial:
𝑝 𝑥 = 𝑥 4 + 𝑥 3 − 17𝑥 2 −21𝑥+36
± 𝑝 𝑞 =± 1, 2, 3, 4, 6, 9, 12, 18, 36 1
Possible Roots: ±1, ±2, ±3, ±4,±6, ±9, ±12, ±18, or±36

6. Possible Roots: ±1, ± 1 3 , ±2,± 2 3 ,±3, ±4,±6, ±9, ±12, ±18, or±36
Example:
Determine the possible list of rational roots for the following polynomial:
𝑝 𝑥 = 3𝑥 3 +9 𝑥 2 −12𝑥−36
± 𝑝 𝑞 =± 1, 2, 3, 4, 6, 9, 12, 18, 36 1,3
Possible Roots: ±1, ± 1 3 , ±2,± 2 3 ,±3, ±4,±6, ±9, ±12, ±18, or±36

7. Possible Roots: ±1, ±2,±3, ±4,±6, ±12,
Example:
Determine the factored form for the following polynomial:
𝑝 𝑥 = 𝑥 4 + 4𝑥 3 − 𝑥 2 −16𝑥−12
± 𝑝 𝑞 =± 1, 2, 3, 4, 6, 12 1
Possible Roots: ±1, ±2,±3, ±4,±6, ±12,
𝑝 𝑥 =(𝑥−2)(𝑥+2)(𝑥+1)(𝑥+3)

8. Example: 𝑝 𝑥 = 𝑥 3 −7𝑥+6 ± 𝑝 𝑞 =± 1, 2, 3, 6 1
Determine the factored form for the following polynomial:
𝑝 𝑥 = 𝑥 3 −7𝑥+6
± 𝑝 𝑞 =± 1, 2, 3, 6 1
Possible Roots: ±1, ±2,±3,±6
𝑝 𝑥 =(𝑥−2)(𝑥−1)(𝑥+3)

9. For Tonight:
Continue to work on the HW from Th.