Question

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

Answer 1

Answer:

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

Step-by-step 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 $\frac{P}{Q}$.

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).