Assignment 21 The Rational Root Theorem (2A.3.4a) Homework: Pg.94 (1-8) Warm Up Solve a. 𝑥! − 3𝑥 = 28 b. 𝑥! + 6𝑥 + 32 = 25

Notes (𝑥 + 3)(𝑥! + 4𝑥 − 1) can be rewritten as 𝑥! + 7𝑥! + 11𝑥 − 3 if you multiply the factors out. If you were to factor 𝑥! + 2𝑥! − 5𝑥 − 6, which of the following could be possible factors: (𝑥 − 3) 𝑥 + 3                  (𝑥 − 7) (𝑥 − 4) (𝑥 − 1) (𝑥 − 6) 𝑥 + 12                  (𝑥 − 2) (𝑥 − 5) (𝑥 + 1) Be able to explain why. Rational Root Theorem If !

! is a rational root of the polynomial 𝑎!𝑥! +⋯+ 𝑎! = 0

with integer coefficients, then p must be a factor of 𝑎! and q must be a factor of 𝑎!. Finding the roots of higher degree polynomials: 1) Find p (factors of the constant) 2) Find q (factors of the leading coefficient) 3) List all the possible rational roots: ± !

!

4) Use synthetic division to find the roots (remainder = 0) 5) Factor down to the quadratic, then solve

Ex.1) Find all the roots of each polynomial a. 𝑥! + 2𝑥! − 5𝑥 − 6 = 0 b. 3𝑥! + 𝑥! − 10𝑥! + 2𝑥 + 4 = 0 *Use the graph to find the roots but show all of the work. Student Workbook Practice (1-8)