# List the potential rational zeros of the following function. Please explain. `f(x)=-4x^4+3x^2-2x+6`

txmedteach | Certified Educator

To find the possible rational zeros of a function, you need to use the rational root theorem. This theorem states that each rational root will be of the form:

`x = +- p/q`

where `p` is an integer factor of the constant term of the polynomial, and `q` is an integer factor of the coefficent of the highest power of `x.`

So, we can easily get our possible p's and q's by factoring 6 and -4, respectively:

`p in {1, 2, 3, 6}`

`q in {1, 2, 4}`

Notice that we're not worried about negatives here, because that is already covered in the form of the root we gave above. Also, this particular form for p's and q's I is different than what I wrote on your other answers to show a new notation. It means the same as what I showed you before!

So, now we can find each possible combination of p's and q's to give us each possible rational root, x:

`x = +- 1/1, +-1/2, +-1/4, +-2/1, +-2/2, +-2/4, +-3/1, +-3/2, +-3/4, +-6/1, +-6/2, +-6/4`

That's a pretty long list, but notice that a lot of these are repeats. For example, `1/1` and `2/2` are the same thing! So, if we simplify the fractions, we can actually reduce the above list to the following possible rational solutions:

`x = +-1, +-1/2, +-1/4, +-2, +-3, +-3/2, +-3/4, +-6`

Most of the time, we like to order these, so let's just do that now:

`x = +-1/4, +-1/2, +-3/4, +-1, +-3/2, +-2 ,+-3, +-6`

And these are your possible rational roots! There are no other possibilities for rational roots, based on the theorem.

I hope that helps!