What are all the possible rational zeros for #y=x^3+x^2-9x+7# and how do you find all zeros?
1 Answer
Explanation:
By the rational roots theorem, any rational zeros of
That means that the only possible rational zeros are:
#+-1, +-7#
Since the sum of the coefficients of
#x^3+x^2-9x+7 = (x-1)(x^2+2x-7)#
We can factor the remaining quadratic by completing the square and using the difference of squares identity:
#a^2-b^2 = (a-b)(a+b)#
as follows:
#x^2+2x-7 = x^2+2x+1-8#
#color(white)(x^2+2x-7) = (x+1)^2-(2sqrt(2))^2#
#color(white)(x^2+2x-7) = ((x+1)-2sqrt(2))((x+1)+2sqrt(2))#
#color(white)(x^2+2x-7) = (x+1-2sqrt(2))(x+1+2sqrt(2))#
Hence the remaining zeros are:
#x = -1+-2sqrt(2)#