Find all the zeros of the polynomial function.

Question

Find all the zeros of the polynomial function. Use the Rational Zero​ Theorem, Descartes's Rule of​ Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero

f(x)= x^3-3x^2-6x+8

Answer 1

Just try a couple simple integers, such as x=1

f(1) = 1^3 -3(1)^2 -6(1) + 8

f(1) = 1-3-6+8 = 0

x=1 is one zero

divide x-1 into the 3 degree polynomial to get a 2nd degree polynomial (a quadratic)

use either long division or synthetic division to get x^2 -2x + 8

which can be factored into

(x-4)(x+2)

which makes

x = 4 or -2

the 3 zeroes are 1, 4, and -2