What are all the possible rational zeros for #f(x)=2x^3+x^2-2x-1# and how do you find all zeros?

1 Answer
Mar 27, 2017

Answer:

All of the roots are rational: #-1/2#, #-1#, and #1#.

Explanation:

Usually, it is hard to find the zeros of third-degree polynomials. However, for this one, we can do so by a special trick.

We can rewrite this polynomial as #2x^3+x^2-2x-1=x^2(2x+1)-(2x+1)#. We can then factor #2x+1# from this, getting #(2x+1)(x^2-1)#. We recall that #a^2-b^2=(a+b)(a-b)# to further factor #(x^2-1)# into #(x+1)(x-1)#. This polynomial then becomes #(2x+1)(x+1)(x-1)#. The roots are therefore #-1/2#, #-1#, and #1#. All of these are rational.