Question

What is the completely factored form of f(x)=x3-2x2-5x+6?

Answer 1

Answer:

f(x) = (x -1)(x + 2)(x -3) is the factored form.

Step-by-step explanation:

The given function is f(x) = x³ -2x² - 5x + 6

We have to write it in the completely factored form.

For this we will use the rational roots theorem first.

In the equation x³ - 2x² - 5x + 6 = 0

P = ± multiples of constant term 6 = ± 1, ±2, ±3, ±6

Q = ± multiples of the coefficient of highest degree term = ±1

By theorem factors will be P/Q

Possible rational roots will be 1, ±2, ±3, ±6

Therefore 1 is a confirm rational root. Now we will find the depressed polynomial from synthetic division to find the other rational roots.

1 | 1  -2  -5   6

        1   -1   -6

-------------------------

   1   -1   -6   0    

So the depressed polynomial will be  (x² - x - 6).

Now can easily factorize this polynomial to get the rational roots.

x² - x - 6 = x² - 3x + 2x - 6

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

Therefore whole factorized form of the polynomial function will be

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

Answer 2

The answer is (x+2)(x-3)(x-1). 
To get this answer first find the possible rational roots using the rational root theorem. Then with one of the potential roots, use synthetic division to divide the polynomial. (If one possible root doesn't work use another). Then simplify.