# How do you use the rational root theorem to list all possible roots for #2x^3+5x^2+4x+1=0#?

##### 1 Answer

The "possible" rational roots are:

The actual roots are:

#### Explanation:

Given:

#f(x) = 2x^3+5x^2+4x+1#

Note that the rational root theorem will only tell us about *rational* zeros, not necessarily all zeros.

By the rational root theorem, any rational zeros of

That means that the only possible *rational* zeros are:

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

In this particular example note that all of the coefficients are positive. Hence

#f(-1) = -2+5-4+1 = 0#

So

#2x^3+5x^2+4x+1 = (x+1)(2x^2+3x+1)#

Note that

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

So the third zero is

So in this particular example all of the roots were rational and therefore findable with the help of the rational roots theorem.