Find the x values

Question

Use the rational root theorem and the factor theorem to help solve the equation. Be sure that the number of solutions for the equation agrees with the property below, taking into account multiplicity of solutions. (Objectives 1 and 2. Enter your answers as a comma-separated list. If a solution has multiplicity p, enter it p times.)
A polynomial equation of degree n has n solutions, and any solution of multiplicity p is counted p times.
16x4 + 24x3 + x2 − 6x + 1 = 0

Answer 1

16x^4+24x^3+x^2-6x+1=0

I always start with +1 or -1.

-1 is a root

therefore x+1 is a factor

use synthetic division or divide to get 16x^3+8x^2-7x+1

-1 is a root again

so x+1 is a factor

divide or use synthetic division to get 16x^2-8x+1 which is a square trinomial

it factors into (4x-1)(4x-1)

therefore 4x-1=0, 4x=1, and x=1/4

the roots are -1, -1, 1/4, and 1/4