What are all the possible rational zeros for #f(x)=x^3-4x^2-3x+14# and how do you find all zeros?
1 Answer
Aug 18, 2016
Explanation:
#f(x) = x^3-4x^2-3x+14#
By the rational roots theorem, any rational zeros of
That means that the only possible rational zeros are:
#+-1, +-2, +-7, +-14#
Trying each in turn, we find:
#f(2) = 8-4(4)-3(2)+14 = 8-16-6+14 = 0#
So
#x^3-4x^2-3x+14 = (x-2)(x^2-2x-7)#
We can factor the remaining quadratic by completing the square:
#x^2-2x-7#
#=x^2-2x+1-8#
#=(x-1)^2-(2sqrt(2))^2#
#=((x-1)-2sqrt(2))((x-1)+2sqrt(2))#
#=(x-(1+2sqrt(2))(x-(1-2sqrt(2)))#
Hence zeros:
#x = 1+-2sqrt(2)#