# 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#