Question

What is the completley factored form of f(x)=x^3-2x^2-5x+6

Answer 1

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

Explanation:

We are given the following function and we are to factorize it completely:

`f ( x ) = x ^ 3 - 2 x ^ 2 - 5 x + 6`

To factorize this completely, we will use the rational roots theorem.

`x ^ 3 - 2 x ^ 2 - 5 x + 6`

P = ± multiples of constant term
`6 = \pm1, \pm2, \pm3, \pm6`

Q = ± multiples of the coefficient of highest degree term
`= \pm1`

So the factors will be
`(P)/(Q)`.

The possible rational roots are
`1, \pm2, \pm3, \pm6`.

1 is a confirmed root and now we will use synthetic division to find the other rational roots:

1 | 1 -2 -5 6

1 -1 -6

___________

1 -1 -6 0

So the polynomial will be
`(x^2 - x - 6)` which can we factorize now.

`x^2 - x - 6 = x^2 - 3x + 2x - 6`

`x(x - 3) + 2(x - 3) = (x+2)(x-3)`

Therefore, the completely factorized form of the given function is f(x) = (x - 1)(x + 2)(x - 3).