# How do you use the rational root theorem to find the roots of #f(x)=x^3+x^2-8x-6 #?

##### 1 Answer

The zeros of

#### Explanation:

Given:

#f(x) = x^3+x^2-8x-6#

The rational root theorem tells us that any rational zero of

That means that the only possible rational zeros are:

#+-1, +-2, +-3, +-6#

Trying each of these in turn, we eventually find:

#f(-3) = (color(blue)(-3))^3+(color(blue)(-3))^2-8(color(blue)(-3))-6 = -27+9+24-6 = 0#

So

#x^3+x^2-8x-6 = (x+3)(x^2-2x-2)#

We can find the zeros of the remaining quadratic by completing the square and using the difference of squares identity:

#A^2-B^2 = (A-B)(A+B)#

with

#x^2-2x-2 = x^2-2x+1-3#

#color(white)(x^2-2x-2) = (x-1)^2-(sqrt(3))^2#

#color(white)(x^2-2x-2) = ((x-1)-sqrt(3))((x-1)+sqrt(3))#

#color(white)(x^2-2x-2) = (x-1-sqrt(3))(x-1+sqrt(3))#

Hence the other two zeros are:

#x = 1+-sqrt(3)#