What are all the possible rational zeros for #f(x)=x^3+x^2-8x-6# and how do you find all zeros?
1 Answer
Possible rational zeros:
Actual zeros:
Explanation:
Given:
#f(x) = x^3+x^2-8x-6#
By the rational roots theorem, any rational zeros of
That means that the only possible rational zeros are:
#+-1# ,#+-2# ,#+-3# ,#+-6#
Trying each in turn, we find:
#f(-3) = -27+9+24-6 = 0#
So
#x^3+x^2-8x-6 = (x+3)(x^2-2x-2)#
We can factor the remaining quadratic by completing the square:
#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 of our cubic function are:
#x = 1+-sqrt(3)#