# Determine the completely factored form of #f(x) = 12x^3 - 44x^2 + 49x - 15#?

##### 1 Answer

#### Answer:

#### Explanation:

By the rational roots theorem, any *rational* zero of

In addition, note that the pattern of signs of coefficients of

Hence the only possible *rational* zeros are:

#1/12, 1/6, 1/4, 1/3, 5/12, 1/2, 3/4, 5/6, 1, 5/4, 3/2, 5/3, 5/2, 3, 5, 15#

We could simply try each of these in turn, but if you are allowed, we can speed up the process of finding the zeros as follows:

Look at the derivative and find out where it is

#f'(x) = 36x^2-88x+49#

#color(white)(f'(x)) = (6x-22/3)^2-484/9+49#

#color(white)(f'(x)) = (6x-22/3)^2-(sqrt(43)/3)^2#

#color(white)(f'(x)) = (6x-22/3-sqrt(43)/3)(6x-22/3+sqrt(43)/3)#

Hence zero at

#sqrt(43) ~~ 6.5#

So the local maximum and minimum are approximately at:

#1/18(22-6.5) = 31/36 ~~ 5/6# and#1/18(22+6.5) = 19/12 ~~ 3/2#

#f(5/6) = 20/9#

#f(3/2) = 0#

Hmmm - a zero near where we expect the minimum. Let's look at the other rational possibility nearby...

#f(5/3) = 0#

So

Looking at the coefficient of the leading term and constant term, the remaining factor (which must be rational) must be

So:

#12x^3-44x^2+49x-15 = (2x-3)(3x-5)(2x-1)#

graph{12x^3-44x^2+49x-15 [-0.448, 2.052, -1.26, 2.49]}