P(x)=x^3-2x^2+x-2?

Question

Factor the polynomial completely and find all its zeros:

Answer 1

Factoring

x^3-2x^2+x-2
x^3 -x - 2x^2 -2

x(x^2-1) - 2(x^2-1)

Factor out x^2-1
(x^2-1)(x-2)

(x-1)(x+1)(x-2)

For P(x) = 0

x=1, x= -1, x=2
.
Q.E.D

Answer 2

first set the polynomial equal to 2; this becomes x^3-2x^2+x = 2; then factor out an x and the equation becomes x(x^2-2x+1) = 2. Then solve x^2-2x+1 which becomes x(x-1)(x-1)=2. The zeroes are x = 2, 1

Answer 3

Use synthetic division:

So let's try 1.

1 ] 3 -2 1 -2
3 1 2
3 1 2 [0

Now we are left with 3x^2+x+2=0, but I don't think it's factorable so I use the quadratic formula.

-1 +/- sqrt (1-4(3)(2)) / 2(3)

-1 +/- sqrt (1-24) / 6

-1 +/- sqrt (-23) / 6

-1 +/- [i sqrt (23) /6]

I think that's the answer.