How do you find the number of possible positive real zeros and negative zeros then determine the rational zeros given #f(x)=x^4+2x^3-9x^2-2x+8#?
1 Answer
The zeros of
#1, -1, -4, 2#
Explanation:
Given:
#f(x) = x^4+2x^3-9x^2-2x+8#
Note that the pattern of the signs of the coefficients is
The pattern of the signs of
By the rational roots theorem, any rational zeros of
That means that the only possible rational zeros are:
#+-1# ,#+-2# ,#+-4# ,#+-8#
In addition, note that the sum of the coefficients of
#1+2-9-2-8 = 0#
Hence
#x^4+2x^3-9x^2-2x+8 = (x-1)(x^3+3x^2-6x-8)#
Note that
#-1+3+6-8 = 0#
So
#x^3+3x^2-6x-8 = (x+1)(x^2+2x-8)#
Finally, note that
#x^2+2x-8 = (x+4)(x-2)#
So the zeros of
#1, -1, -4, 2#