Setup and find all roots of the polynomial algebraically​ (Not Graphing)​ :

x cubed minus two x squared minus​ thirty-five x equals zero
Understand​ & Think: Do you understand what is being asked and what you need to​ do? Do you have a list of things you​ know, ideas to​ try, and a plan for doing​ it? Identify the problem and list your strategies.

1 answer

The problem is to find all the roots of the polynomial algebraically. To do this, I will use the factor theorem to factor the polynomial into linear factors. Then, I will set each factor equal to zero and solve for the roots.

Solution:

Factor the polynomial:
x^3 - 2x^2 - 35x = (x - 5)(x^2 + 5x + 7) = 0

Set each factor equal to zero and solve for the roots:
x - 5 = 0
x = 5

x^2 + 5x + 7 = 0
(x + 7)(x + 1) = 0
x = -7
x = -1

The roots of the polynomial are x = 5, -7, and -1.