# What are all the possible rational zeros for #y=2x^5-x^4-26x^3+13x^2+72x-36# and how do you find all zeros?

##### 1 Answer

The "possible" rational zeros are:

#+-1/2, +-1, +-3/2, +-2, +-3, +-4, +-9/2, +-6, +-9, +-12, +-18, +-36#

The actual zeros are:

#+-3# ,#+-2# ,#1/2#

#### Explanation:

By the rational root theorem, any rational zeros of

That means that the only possible rational zeros are:

#+-1/2, +-1, +-3/2, +-2, +-3, +-4, +-9/2, +-6, +-9, +-12, +-18, +-36#

In addition, note the the ratio of the first and second terms is identical to that of the third and fourth terms and the fifth and sixth terms. So this quintic will factor by grouping:

#2x^5-x^4-26x^3+13x^2+72x-36#

#= (2x^5-x^4)-(26x^3-13x^2)+(72x-36)#

#= x^4(2x-1)-13x^2(2x-1)+36(2x-1)#

#= (x^4-13x^2+36)(2x-1)#

We can factor the remaining quartic as a quadratic an

#= ((x^2)^2-13(x^2)+36)(2x-1)#

#= (x^2-9)(x^2-4)(2x-1)#

#= (x^2-3^2)(x^2-2^2)(2x-1)#

#= (x-3)(x+3)(x-2)(x+2)(2x-1)#

So the zeros are:

#+-3# ,#+-2# ,#1/2#