Which linear expression is a factor of the polynomial function f (x) = x3 + 2x2 - x - 2?

The function

f (x) = x3 + 2x2 - x - 2

is a trinomial and it is expected that it has three factors corresponding to the three roots. The possible roots could 1, - 1, 2, or - 2. Trying out 1

f (1) = 1 + 2 - 1 - 2 = 0

According to the remainder theorem (x - 1) is a factor. By doing synthetic division

1 | 1 2 - 1 - 2

1 3 2

1 3 2 0

f (x) = (x - 1) (x2 + 3x + 2)

which can further be factored into

f (x) = (x - 1) (x + 1) (x + 2)

The three factors are

x-1

x+1

x+2

which are all linear expressions