What are all the possible rational zeros for #f(x)=5x^3-9x^2-12x-2# and how do you find all zeros?
1 Answer
The zeros of
#-1/5" "# and#" "1+-sqrt(3)#
Explanation:
Given:
#f(x) = 5x^3-9x^2-12x-2#
By the rational root theorem, any rational zeros of
That means that the only possible rational zeros are:
#+-1/5, +-2/5, +-1, +-2#
First check
#f(1) = 5-9-12-2 = -18#
#f(-1) = -5-9+12-2 = -4#
We can deduce that not all of the zeros are rational, since there are not enough factors to go around.
Trying the other possibilities, we eventually find:
#f(-1/5) = -5(1/125)+9(1/25)+12(1/5)-2#
#color(white)(f(-1/5)) = (-1-9+60-50)/25 = 0#
So
#5x^3-9x^2-12x-2 = (5x+1)(x^2-2x-2)#
#color(white)(5x^3-9x^2-12x-2) = (5x+1)(x^2-2x+1-3)#
#color(white)(5x^3-9x^2-12x-2) = (5x+1)((x-1)^2-(sqrt(3))^2)#
#color(white)(5x^3-9x^2-12x-2) = (5x+1)(x-1-sqrt(3))(x-1+sqrt(3))#
So the zeros are:
#-1/5" "# and#" "1+-sqrt(3)#