# How do you use the rational roots theorem to find all possible zeros of #P(x) = 12x^4 + x^3 + 4x^2 + 7x + 8 #?

##### 1 Answer

Use the rational roots theorem to find possible candidate *rational* zeros, any thereby find that it has none.

#### Explanation:

By the rational roots theorem, any *rational* zeros of

So in our example, it means that the only possible *rational* zeros are:

#+-1/12# ,#+-1/6# ,#+-1/4# ,#+-1/3# ,#+-1/2# ,#+-2/3# ,#+-1# ,#+-4/3# ,#+-2# ,#+-8/3# ,#+-4# ,#+-8#

That's rather a lot of possible zeros to try, but you can narrow it down a little by noting that the coefficients of *rational* zeros:

#-1/12# ,#-1/6# ,#-1/4# ,#-1/3# ,#-1/2# ,#-2/3# ,#-1# ,#-4/3# ,#-2# ,#-8/3# ,#-4# ,#-8#

Substituting each of these for *rational* zeros.

That's all the rational roots theorem tells us.

In fact, this particular quartic only has non-Real Complex zeros.